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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Overview
Chapter I expand the students’ ability to use polynomials to represent and solve
problems reflecting real-world situations while focusing on symbolic and
graphical patterns.
The study of the properties and graphs of polynomial functions is useful to
scientists, astronomers, physicists and chemists in the field of scientific research.
These properties are useful in making satellite dishes, car headlights, radio
telescopes and reflecting telescopes.
The ideas and skills learned in this chapter will help the students organize
information, interpret and solve problems logically. They will also enable the
students to come up critical evaluations of the solutions found.
Throughout the chapter, interesting problem contexts serve as the foundation for
instruction. As lessons unfold around these problem situations, classroom
instruction tends to follow a common pattern as elaborated upon in the
instructional procedure.
Classroom activities are designed to actively engage students in problem
investigation and making sense of problem situations. They will work together
collaboratively in heterogeneous groupings: in pairs or in small groups. They will
communicate their mathematical thinking and the results of their group efforts.
Focus Questions
1. How can polynomial equations be used to provide accurate models
of practical problems that involve three dimensions?
2. How can a polynomial model be used to solve problems where
maxima or minima are of significant importance?
3. How can polynomial expressions be used to represent and predict
social or fiscal changes over time?
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Objectives
BEC Standards
After completing Chapter 1, the students should be able to:
 identify a polynomial function from a given set of relations;
 determine the degree of a given polynomial function;
 find the quotient of polynomials by using the division algorithm and
synthetic division;
 find the quotient using synthetic division and the Remainder
Theorem when p(x) is divided by (x-c);
 state and illustrate the Remainder Theorem;
 find the value of p(x) for x = k
Remainder Theorem;
by synthetic division and the
 state and illustrate the Factor Theorem;
 find the zeroes of polynomial functions of degree greater than two
by using:




Factor Theorem
Factoring
Synthetic Division
Depressed Equations
 draw the graph of polynomial functions of degree greater than two
(use a graphing calculator, if available).
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Key Terms
Terms
when numbers are added or subtracted, they are called
terms. Example: 4x² + 7x − 8 is a sum of three terms.
Factors
when numbers are multiplied, they are called factors.
Example: (x + 1)(x + 2)(x + 3) is a product of three factors.
Variable
is a symbol that takes on values.
Value
is a number; thus if x is the variable and has the value 4,
then 5x + 1 has the value 21.
Constant
is a symbol that has a single value.
Example: The symbols '5' and ' ' are constants.
The beginning letters of the alphabet a, b, c, etc. are typically
used to denote constants, while the letters x, y, z, are
typically used to denote variables.
Example: if we write y = ax² + bx + c, we mean that a, b, c
are constants (i.e. fixed numbers), and that x and y are
variables.
Monomial in x
is a single term of the form axn, where a is a real number and
n is a whole number. Examples: 5x3, −6.3x, 2.
Polynomial in x
is a sum of monomials in x. Example: 5x3 − 4x² + 7x − 8
Degree of a Term is the sum of the exponents of all the variables in that term.
In functions of a single variable, the degree of a term is
simply the exponent. Example: The term 5x³ is of degree 3 in
the variable x.
Leading Term
of a Polynomial is the term of highest degree. Example: The
leading term of this polynomial 5x³ − 4x² + 7x − 8 is 5x³.
Leading Coefficient of a Polynomial is the coefficient of the leading term.
Example: the leading coefficient of that polynomial is 5.
Degree of a Polynomial is the degree of the leading term. Example: the degree
of this polynomial 5x³ − 4x² + 7x − 8 is 3.
Constant Term
of a Polynomial is the term of degree 0; it is the term in
which the variable does not appear. Example: The constant
term of this polynomial 5x³ − 4x² + 7x − 8 is −8.
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
General Form
of a Polynomial shows the terms of all possible degree.
Example, is the general form of a polynomial of the third
degree: ax³ + bx² + cx + d. Notice that there are four
constants: a, b, c, d.
Polynomial function has the form: y = A polynomial
A polynomial function of the first degree, such as
y = 2x + 1, is called a linear function; while a polynomial
function of the second degree, such as y = x² + 3x − 2, is
called a quadratic.
Domain and range The natural domain of any polynomial function is:
−  < x <  . x may take on any real value on the x-axis.
The domain of a function is the set of values of the
independent variable, which are the values of x.
The range of a function is the set of values of the
dependent variable, which are the values of y.
Polynomial equation is a polynomial set equal to 0. P(x) = 0.
Example. P(x) = 5x³ − 4x² + 7x − 8 = 0
Root, or Zero, of a polynomial is a solution to the polynomial equation,
P(x) = 0. That is, the number r is a root of a polynomial P(x)
if and only if P(r) = 0.
Remainder Theorem if a polynomial P(x) is divided by a linear function x - c,
the remainder is P(c). The degree of the remainder is
always one less than the degree of the divisor.
Factor Theorem
If x − c is a factor of a polynomial P(x), then P(x) = 0.
Synthetic division is a shortcut method of doing long division of polynomials
when the divisor is of the form x + c.
Rational Root Theorem - In the polynomial, an xn + an-1 xn-1 + · · · + a1 x + a0 = 0
where the coefficients an ,an-1 , . . . ,a1 ,a0 are integers.
If a rational number c/d, is factored to its lowest terms, is a
solution to the equation, then c is a factor of the constant
term of the polynomial and d is a factor of the leading
coefficient of the polynomial.
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Concept Map
Addition
are performed and
expressed using
are
explained
through
Algebraic
Expressions
in the
form of
POLYNOMIALS
Evaluation of
polynomials
by substitution
Subtraction
Multiplication
Division
Operations on
polynomials
can be simplified
through
Terms and
degrees of
polynomials
Polynomial functions
P(x) = AnXn+An-1Xn-1+…+ A1X+A0
Algorithm
leads to
can be
illustrated by
Graphing of polynomial
functions
Zeroes of polynomial
functions
Rational zeroes
Polynomial inequalities
5
Remainder
Theorem
Factor
Theorem
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Lesson
1
RECOGNIZING POLYNOMIALS
TIME
1 session
SETTING
Computer room and Math room
OBJECTIVES
At the end of this lesson, the students should be able to:
 tell whether an expression is a polynomial or a non-polynomial
function;
 identify polynomials from a given number of algebraic expressions;
and
 state the characteristics of polynomials.
PREREQUISITE
Students should have learned the following concepts and their
definitions:
1. algebraic expressions
5. monomial
2. term
6. binomial
3. coefficient
7. trinomial
4. polynomials
8. degree of polynomial
RESOURCES
 chart
 Manila paper
 marker pen
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
PROCEDURE
Opening Activity: POLYNOMIAL QUIZ BEE
A. Begin with a review of algebraic expressions. Ask volunteers to
form two groups. Acting as contestants, they will answer the three
questions asked by the teacher. For the first set, the contestants
will translate the following mathematical expressions to phrases.
1. 3x + 4
2. 6x2
(Answer: 4 is added to thrice a number)
(Answer: a number, squared, is multiplied by 6)
2x
(Answer: a number is multiplied by 2/3, or twice a
3
number is divided by 3)
3. 3.
B. For the second set of questions, students will translate the following
English phrases to mathematical expressions.
1. A number x is raised to the 2/3 power. (Answer: x2/3)
2. Six times a number x added to the product of three and the cube
of (Answer: 6x + 3c3)
3. The square root of x subtracted from the product of three and
x. (Answer: 3x - x )
 3 
4. Three divided by the product of 4 and X. Answer:  
 4x 
C. For the third set of questions, students will express the following
using nonnegative exponents.
a. a-2
b. ab-1
c. 2x-4
d.
2 2 -3
ab
3
D. Process the activity by asking:
 Why do we have to translate mathematical symbols to English
and vice versa?
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
It is easier to solve word problems if we know how to take the
English problem and translate it into mathematical statements
which are also known as algebraic expressions.
 How can algebraic expressions be used in understanding
polynomials?
A polynomial is basically an algebraic expression that consists
of a sum of terms, each term being the product of a constant
and a nonnegative (or zero) power of a variable or variables.
For example: 3x3 - 2x + 0.5x2 + 6
Main Activity
A. Given the students’ basic understanding of polynomials, ask them
to identify the polynomials in the following questions:
3x
3
and
?
4x
4
2. Which of the two is a polynomial?
1. What is the difference between
B. Ask the students to give examples of mathematical expressions
and let them determine whether the given examples are
polynomials or non-polynomials.
Discussion Ideas
Considering the translated mathematical symbols in our review and
given the set of algebraic expressions below, ask the students to
answer the following:
Group I
a.) 6x + 3c2
b.) 6x2
c.) 3x + 4
3x
d.)
4
Group II
a.) 3x-3
b.) 6x1/2
c.) 3x + x
3
d.)
4x
1. What can be said about the exponents of the variables in the first
group of examples? The exponents in the second group of
examples?
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
2. Compare example b in both groups. How do they differ?
3. Look at the second term in example c in both groups. How are they
different from each other?
4. Where are the variables in example d in both groups located?
5. What characteristics of the terms in Group I make them
polynomials?
6. Are all polynomials algebraic expressions? Why or why not?
7. Are all algebraic expressions polynomials? Why or why not?
Summarize the responses and formulate the key learning points on
polynomials.
Key Learning Points
 A polynomial is an algebraic expression involving only
nonnegative-integer powers of one or more variables and
containing no variable in the denominator.
 Polynomial functions are functions with x as an input variable. They
are made up of several terms, each of which consists of two
factors, the first being a real number coefficient, and the second
being x (or any variable) raised to some nonnegative integer
power.
 Polynomial functions are functions of the form: f(x) = anxn + an-1xn-1
+ ... + a1x + a0.
 The value of n must be a nonnegative integer. That is, it must be
a whole number equal to zero or a positive integer.
 The numerical coefficients of the terms in the polynomial
function are an, an-1,..., a1, a0. They are real numbers.
 The degree of the polynomial function is the highest value for n
where an is not equal to 0.
Extension Ideas
1) Which of the following are polynomials? Give the reason for your
answer.
a2  a
a.) 2x3
d.) 2
a 2
b.) 7 – 3a2
e.) 5x2 – 2xy + 3y2
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
c.) x3 + x-2
f.) 3x +
2
(Answers: a, b, e are polynomials. The rest have variables with
negative or non-integer powers)
2) Based on the definition given previously, why is the expression
1
x2
not a polynomial?
(Answer: It is not a polynomial because
1
x
2
is equal to x–2, which is
a negative power of the variable x.)
Closing Activity
Ask the students to use the words “term” and “degree” in two
sentences each. In one sentence, the mathematical meaning of the
word should be used. In the other sentence, the non-mathematical
meaning of the word should be used.
Example:
The term 5x means five multiplied by x; meanwhile, x3 means x to the
third power.
ASSESSMENT
Which of the following expressions is a polynomial? Write P before the
number if the expression is a polynomial and NP if it is a nonpolynomial.
1. 4x5 – 3x4 + 2x3 – x2 + 2x5 + 6
2. 6x + 2x + 5 + 3x
3. 2/3 x7 – x2 + 8
4. 10x2 + 3x – 4x – 7
5. 3x4 + x1/3 + 1
HOMEWORK
State whether each of the given expressions is a polynomial. Write P if
the expression is a polynomial and NP if it is a non-polynomial.
1. 6x + 3x3
6. x-2
2. x1/2
7. x/2
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
3. x
8. 5x - 3
4. 3x/4
9. 8 – 5b2
5. 2/4x
10. 12x4
REFERENCES
Dalton, Leroy C. Using Algebra
Foster, Alan G. Algebra 2 With Trigonometry. Applications and
Connections.
Jose-Dilao, Soledad. Advanced Algebra, Trigonometry and Statistics.
Functional Approach. 65.
Travers, Kenneth J. Algebra 2 with Trigonometry.
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Lesson
2
TERMS AND DEGREES OF A POLYNOMIAL
TIME
1 session
SETTING
Math room
OBJECTIVES
At the end of this lesson, the students should be able to:
 classify polynomials according to their number of terms; and
 determine the degree of the polynomial.
PREREQUISITE
Students are expected to have understood the concept of polynomials.
They should have imbibed the skill of identifying polynomials from a list
of algebraic expressions.
RESOURCES
 drill board or flash cards
 chart
PROCEDURE
Opening Activity
A. Conduct a review on polynomials by asking: When is an expression
a polynomial?
B. Ask the students to present and discuss their assignments in
Lesson 1. As an alternative activity, test the students’ skill in
identifying polynomials. Using flash cards, ask the students to tell
whether each of the given algebraic expressions is a polynomial or
not.
2
1. 6x + 3x3
5.
4x
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
2. x1/2
6. x-2
3. x
7. y2 - 5
4.
3x
4
8.
x
2
C. Introduce the new lesson using a motivational activity. Introduce the
lesson by telling the students that:
It is common practice that names like MACMAC, JUNJUN, and
JETJET are written thus: MAC2, JUN2, JET2. Mathematically
speaking, MAC2 is MACC, where only C is taken twice. What do
(MAC)2 and MAC2 represent?
(Possible Answer: The above examples represent the degree of the
variables and the application of one of the laws of exponents.
(Mac)2 is read as MMAACC while MAC2 is read as MACC.)
Main Activity
Present the lesson by giving three groups of examples. Ask the
students to observe the similarities and differences between the
variables and the exponents. Let them write their observations.
Group A
Group B
Group C
a.
b.
c.
d.
a.
b.
c.
d.
a.
b.
c.
d.
6abc
3x2y2z
equ
5
6a + bc
3x2 + y2z3
eq – gu
5x – xo
a+b+c
3x2 – 3y2 + z2
e5 – g4 – u3
5x2 - 1
Discussion Ideas
A. Discuss the concepts “terms” and “degrees of polynomials” by
asking the following questions:
1. Compare the examples given in each group.
differ?
How do they
2. How many terms are given in each of the following?
a.) group A
b.) group B
c.) group C
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
3. How would you classify the examples in
a.) group A?
b.) group B?
c.) group C?
4. When is a polynomial
a.) a monomial?
b.) a binomial?
c.) a trinomial?
If a polynomial has four or more terms, how is it classified?
5. In group A, what is the highest exponent of the variables? In
group B? In group C?
6. In example c group C, 5 is the highest exponent, therefore the
polynomial is of degree 5. How is the degree of a polynomial
determined?
Extension Ideas
Allow the students to work on the following exercises in order to
practice their skills in identifying the terms and degrees of polynomials.
A. Identify each polynomial according to the number of its terms.
1. x2 + 10 x + 5
2. 3a + 2b
3. 4xyz
4. 2m2n-mn2
5. 7a + b -2
B. Give the degree of each of the following polynomials
1. 3x2 –x2 + x -3
2. x2 – x6 + x4 + 3
3. 5x2y – 4x4y3-2
4. x + 2x2 + 3x2 + 6
5. 3x6 + 6x4 + x2 - x
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Closing Activity
Summarize the key learning points by asking each group to complete
the following phrases:
1. The degree of a term is……..
2. The degree of a polynomial is…….
3. The standard form of a polynomial is arranged in……
4. Polynomials are classified into….
5. The degree of a nonzero constant polynomial is….
6. The zero polynomial does not have any…..
Key Answers
1. the sum of the exponents of the variable.
2. the highest of the degrees of its terms (after it has been
simplified)
3. descending order
4. monomials, binomials, trinomials and multinomials
5. zero (0)
6. degree
ASSESSMENT
Arrange the following polynomials in standard form. Classify each
polynomial according to the number of its terms. Then determine the
degree of each of the following polynomials:
1. 4x2 – x2 + x – 2
2. 3x2 – x6 + x4 + 1
3. 5x2 – x6 – 4x4
4. 5x – 2x2 + 3x2 + 6
5. 6x6 + 6x4 + x2 – x
6. 2x2 + 8x – x10 - 5 + 20x5
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
HOMEWORK
Rewrite each of the following polynomials in descending powers of the
variable and find the degree of each polynomial:
1.
2.
3.
4.
5.
6. 2 – ½ t – 2t2
7. x – 7x3 + x4 – 3x2
8. m2 – 11 - m
9. 6y + 7y2 – 2 – 5y3
10. 13m2
2
-5 – 2x
x4 + 2 – 3x
½ m – ¼ m2
5 – 2p + 6p2
REFERENCES
Cruz, F.R. Expanding Mathematics.
Jose-Dilao. Advanced Algebra, Trigonometry and Statistics. Functional
Approach. 65–66.
Math IV: Advanced Algebra, Trigonometry and Statistics – BEC
Travers, Dalton and Brunner. Using Algebra. Third edition.
17
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Lesson
3
EVALUATING POLYNOMIALS EXPRESSIONS
TIME
1 session
SETTING
Math room
OBJECTIVE
At the end of this lesson, the students should be able to evaluate
polynomial expressions for specified values of the variables.
PREREQUISITE
Students are expected to know the following concepts or to have the
following skills:
 determining the degree of polynomials; and
 classification of polynomials according to the number of terms.
RESOURCES
 drill board or flash cards
 chart
PROCEDURE
Opening Activity
a. Recall the concepts learned the previous day. Discuss the assigned
items. Focus on items that were not accurately done by the
students.
b. Using flash cards, prepare two samples per mathematical
operation.
c. Invite the students to evaluate a set of integers using the four
fundamental operations on integers.
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
d. Sample exercises on integer operations:
Set B
Set A
What is -6 + 19 ?
What is -4 - (16) ?
What is 8 + (-7) ?
What is 12 - (-4) ?
What is -15 + 16 ?
What is -18 - (6) ?
What is 17 + (-3) ?
What is -6 - (-10)?
What is -5 + 9 ?
What is -3 x 3 ?
What is -8 - (-16)?
What is -21 ÷7) ?
What is 7 x 7 ?
What is 40 ÷(-4) ?
What is 1 x 7 ?
What is -42 ÷(-7)?
What is 8 x 3 ?
What is -6 ÷ (6) ?
What is 9 x 9 ?
What is 5 ÷ (1) ?
e. After the short drill, explain to the students that they will apply their
basic knowledge on how to add, subtract, multiply and divide
integers in evaluating polynomial expressions.
Main Activity
1. Relate the concept of polynomials with a simple machine. Ask the
students:
 Do you know what pencils are made of?
 What materials are needed in the manufacture of pencils?
Materials that are needed to produce an output are called
inputs.
 If mathematical symbols/expressions constitute inputs, what
would the output be if we put these symbols in a simple
machine?
2. Post on the board a simple machine which will be used to evaluate
the following expressions.
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Example 1: Evaluate 3x when x = 2
x=2
This machine multiplies
every input(x) by 3
input
3x
output
6
The value of the polynomial function is 6.
Suppose the following values of x were used as inputs in the machine.
What would the corresponding output be?
when x = 3
x = -2
x = -5
3x = 3(3) = 9
3x = 3(-2) = -6
3x = 3(-5) = -15
Example 2: If f(x) = -x3 + 2x2 + x – 3, find the value of the function
when x = 2.
Solution
F(2) = -(2)3 + 2(2)2 + (2) – 3
= -8 + 2(4) + 2 – 3
= -8 + 8 + 2 – 3
= -1
Therefore, the value of the polynomial function when x = 2 is –1.
3. Divide the class into 5 groups. Give each group a strip of Manila paper
where each of the following is written. Let the groups find the value of
f(x) and tell them to write their solutions on the board.
a. f(x) = 2x2 + 3x – 5
where x = -1
b. f(x) = -4x3 – x4 + 3x + 1
x =1
c. f(x) = x3 – 2x2 + 3
x =4
d. f(x) = -3x2 + 5x
x = -2
e. f(x) = -2x + 26
x =0
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Discussion Ideas
Ask the students to answer the following questions:
1. How is a polynomial function f evaluated for a given value of x?
2. What are the steps in evaluating a polynomial function?
3. How are polynomials added or subtracted?
4. How are polynomials multiplied and divided? What laws should be
followed?
Extension Ideas
Seatwork
Ask the students to simplify the expressions below and find their
values when x = -2 and y = 3
a. 4x + 2y + 3(2x-5y)
b. 4x2 – 5xy2 + 3xy (2x-3y)
Closing Activity
Summarize the key learning points as follows:
 To find the value of a polynomial function, substitute the given
value of x and evaluate the resulting numerical expression.
 The value of a polynomial function is the value of the polynomial for
a given value of the variable.
ASSESSMENT
Evaluate the following, given that x = -1, y = 2, z = 3.
1. x2 + xy + y2
2. 2x2 + 3y – z
3.
5 x 2  2y 2
3z
4. 4x4 + 5y3 + y2 + z3
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
HOMEWORK
Evaluate the given polynomials for the specified values of the
variables.
a. -3x2 + 3
for x = -5
b. 4x2 – y
for x = 2 and y = -3
c.
xy2z
–5
for x = 7, y = 3 and z = -5
REFERENCES
Addison and Wesley. Algebra and Trigonometry.
Cruz. Expanding Mathematics.
Jose-Dilao. Advanced Algebra, Trigonometry and Statistics. Functional
Approach. 67–68.
Travers. Algebra 2 With Trigonometry.
Yu-hico. Experiencing Mathematics IV for High School
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Lesson
4
ADDITION AND SUBTRACTION OF POLYNOMIALS
TIME
1 session
SETTING
Math room
OBJECTIVES
At the end of this lesson, the students should be able to:
 add and subtract polynomial expressions, and
 apply some formulas in adding and subtracting polynomial
expressions.
PREREQUISITE
1. Students should have already learned how to evaluate polynomial
expressions.
2. Students should know the different properties of multiplication,
addition, etc.
RESOURCES
 drill board or flash cards
 chart
PROCEDURE
Opening Activity
1. Ask the students to recall the concepts learned the previous day
and to discuss the assigned items.
 Distributive Property
 + (a – b); –(a – b), the Rule of Signs
 How do you simplify grouping symbol/s preceded by a plus
(+)sign? a minus (-) sign?
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
2. Introduce the fundamental operations on polynomials. Ask the
students:
 What methods are used in adding and subtracting polynomial
expressions?
3. Present some illustrative examples for students to analyze until they
have identified the two methods, namely, the horizontal and the
vertical methods.
Example 1: Add 3x2 + 2 and 7x2 – 6x + 3
Solution
Horizontal Method
(3x2 + 2) + (7x2 – 6x + 3)
Vertical Method
3x2
+2
7x2 – 6x + 3
10x2 – 6x + 5
10x2 – 6x + 5
Example 2: Simplify (8x2 – 6x + 7) – (4x2 + 3x – 12)
Solution
Horizontal Method
(8x2 – 6x + 7) – (4x2 + 3x – 12)
Vertical Method
8x2 – 6x + 7
-4x – 3x + 12
4x2– 9x + 19
4x2 – 9x + 19
4. Allow individual students to do some practice exercises. Ask them
to use the two methods in adding and subtracting the following
polynomial expressions.
a) (7x2 – 3y2) + (7x2 + 3y2)
b) (-3n2 + n – 8) + ( 6n2 – 2n – 5)
c) (12y2 + 4y + 7) – (9y2 – 8y – 6)
d) (2m2n2 – 6mn – 13) – (6m2n2 – 3mn – 7)
24
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
5. Provide a lecture-discussion on polynomials in the real world. See
Teacher Notes.
Main Activity
Divide the class into 5 groups. Give each group a strip of Manila paper
wherein all the necessary solutions should be recorded. Instruct the
students to solve the problem accurately by applying the formula given
to them.
Problem 1
Find the surface area of a metal cylinder if r = 12 cm and h = 25 cm.
Use 3.14 for the value of pi.
Problem 2
A room has a perimeter of 44 feet and a height of 8 feet. The room
has three 4 feet by 4 feet windows and one 2 feet by 6 feet door. Find
the number of rolls of 2 feet by 5 feet wallpaper needed to cover up the
computer room.
Discussion Ideas
Ask the students to present their work on the board and to assign one
member per group to discuss the results of their activity.
Extension Ideas
Challenge the students to find the force on an aquarium window that is
2 feet below the water surface if the window is 5 feet wide by 4 feet
high. Use the formula below.
F = 31h2w + 62dhw
Where: d = depth, in feet, of a window below the surface of water
h = height in feet
w = width in feet
Note: F is the force in pounds.
25
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Closing Activity
Synthesize the important points of the lesson by asking the students to
verify if the following statements are true:

To add two polynomials, place a plus sign between them, then
simplify. To simplify, combine all like terms. Then, if possible,
arrange the terms in descending powers of a variable.

The subtraction property, a – b = a + (-b), can be used to subtract
one polynomial from another. In subtracting polynomial
expressions, simply follow the rules on the subtraction of integers
by changing the sign of the subtrahend then proceed to addition.

Distributive Property
ASSESSMENT
Find the sum or difference of the following polynomials:
a.
b.
c.
d.
5x3y – 3 (2x3y + 5x) + 18x – 3
5x + 3 (2x + 4) – 5 (2y + 3) + 18y
3x(2x + 4) – 2x2 + 5 (x – 3)
16xy2z + 4x2yz – (8x2yz + 2xy2z)
HOMEWORK
1) Find the surface area of a metal cylinder whose radius is 18 cm and
whose height is 32 cm.
2)
The computer room has a perimeter of 56 feet and a height of 12 feet.
The room has two 4 feet by 5 feet windows and a 3 feet by 6 feet door. Find the
number of rolls of 2 by 5 feet wallpaper needed to cover up the computer room.
REFERENCES
Cruz. Expanding Mathematics.
Travers. Algebra 2 With Trigonometry.
Dalton, Leroy C. Using Algebra. Third Edition.
Yu-hico. Experiencing Mathematics IV For High Schools.
26
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Teacher Notes
POLYNOMIALS ON THE JOB
Formulas are used in different fields of study. Often the right side of a formula is
a polynomial wherein the addition and subtraction of polynomials are usually
involved. Some formulas in the operation of polynomial expressions have
connections to real-life situations.
Example 1)
A metalworker wants to find the surface area S of a metal container shaped like a
cylinder. (The teacher will show to the class the illustration of a cylinder).
The formula for the surface area of a cylinder is:
S = 2 (3.14) r2 + 2 (3.14) rh,
where:
r
is the radius of the base
h
is the height,

is approximated by 3.14
Example 2)
Wallpaper hangers often use formulas to estimate how much wallpaper is
needed for a room.
Formula: R =
1
ph – (½) n
30
where:
r
is the number of rolls of wallpaper needed
p
is the perimeter of a room in feet
h
is the height of the walls in feet
n
is the number of normal-sized doors and windows.
27
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Lesson
5
MULTIPLICATION OF POLYNOMIALS
TIME
One session
SETTING
Math room
OBJECTIVE
At the end of this lesson, the students should be able to find the
product of two polynomial expressions.
PREREQUISITE
Students should have knowledge of the laws of multiplication of
exponents.
a.) xa . xb = xa + b
b.) ( xa)b = xab
c.) ( xayc)b = xab.ybc
d.) x0 = 1, x  0
RESOURCES




drill board or flash cards
chart
Manila paper
marker pen
PROCEDURE
Opening Activity
A. Have a review on the addition and subtraction of polynomial
expressions and the application of some formulas in the real world.
28
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
B. Introduce the FOIL method for multiplying polynomial expressions
to the students. The FOIL method is an application of the
distributive property that makes the multiplication of polynomials
faster.The product of two binomials is the sum of the products of
the:
irst terms
uter terms
nner terms
ast terms
C. lllustrate how to multiply binomials using the FOIL Method:
Example 1 Use the FOIL method to find (9a – 3) (a + 4)
Solution:
(9a – 3)(a + 4)
= 9a2 + 36a – 3a – 12
= 9a2 + 33a - 12
Example 2 Find (2x2 + 10x – 4)(x – 12)
Solution:
(2x2 + 10x – 4)( x – 12)
= (2x2 + 10x – 4)x – (2x2 + 10x – 4)12 (Distributive
Property)
=
+
– 4x –
– 120x + 48
= 2x3 – 14x2 – 124x + 48
2x3
Main Activity:
10x2
24x2
MULTIPLICATION OF POLYNOMIALS
Divide the class into 5 groups. Give each group one whole Manila
paper and a marker pen. Distribute the Activity Sheets, one for each
group.
Instruction: Find each product.
1. (a + b)(a2 – ab + b2)
6. (2x – 3)(x2 – 3x – 8)
2. (x – y)(x2 + xy + y3)
7. (m – 4)(3m2 + 5m – 4)
3. r(r – 2)(r – 3)
8. (b + 1)(b – 2)(b + 3)
4. (2x – 3)(x + 1)(3x – 2)
9. (2a + 1)(a – 2)2
5. (a – b)(a2 + ab + b2)
10. (2k + 3)(k2 – 7k + 21)
29
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Discussion Ideas
1. Ask the students to present their work on the board and assign one
member per group to discuss the results of the activity.
2. Ask the students: What strategies or steps were applied by your
group in multiplying polynomials?
3. Summarize their responses, then reinforce their ideas by presenting
the following key learning points:
 In the multiplication of polynomials, three important properties of
exponents are useful, namely:
Product of Powers
Let x be any real number, and let a and b be any positive
integers. Then (xa)(xb) = xa+b
Power of a Power
Let x be any real number, and let a and b be any positive
integers. Then (xa)b = xab
Power of a Product
Let x and y be any real numbers, and let a and b be any positive
integers. Then (xy)a = xaya
 The Distributive Property and the FOIL method can be used in
multiplying polynomial expressions.
 In multiplying a polynomial by another polynomial, the vertical
and horizontal methods of finding products are also applicable.
Closing Activity
Ask the students to work in pairs. Let each one give some tips on how
to multiply polynomials the easy way. Let them share their ideas with
their respective partners.
30
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
ASSESSMENT
Find each product.
1. (3x + 2)(5x + 1)
2. (4xy + 5)(3xy – 6)
3. (y – 3z)(y2 + 3yz + 9z2)
4. (x + 1)(x – 1)(x + 1)
5. (x2 + 4xy + 4y2)(x + 2y)
HOMEWORK
Find the area of each figure.
1.
(k2 – 7k + 21)
2.
(2k + 3)
(2a + 4b)
3.
4.
(5x2 – 6x + 10)
(4x – 6)
(2x + 4)
REFERENCES
Cruz. Expanding Mathematics.
Travers. Algebra 2 With Trigonometry.
Dalton, Leroy C. Using Algebra. Third edition.
Yu-hico. Experiencing Mathematics IV for High School.
31
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Student Activity 5
MULTIPLYING POLYNOMIALS
Challenge Problem
 Show geometrically that (a + b)2 = a2 + 2ab + b2
Solution
a. Draw the diagram of a square that has sides of length a + b.
a
b
b
b
a
a
a
b
b. The area of the square can be expressed as (a + b)2 or by a2 + 2ab + b2.
32
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Lesson
6
DIVISION OF POLYNOMIALS
TIME
One session
SETTING
Math room
OBJECTIVE
At the end of this lesson, the students should be able to find the
quotient when a polynomial in x is divided by x – c, using long division.
PREREQUISITE
Students should have gained knowledge of the multiplication of
polynomial expressions by applying the three properties of exponents.
RESOURCES




drill board or flash cards
chart
Manila paper
marker pen
PROCEDURE
Opening Activity
A. Using flashcards, drill the students on the four fundamental
operations on integers.
Review them on the multiplication of polynomials using the FOIL
Method and the Distributive Property. Likewise, check the students’
assignments.
33
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
B. Begin the new lesson with a short activity named “Number Puzzle”.
Group the students by threes and let them answer the cross
number puzzle. Allow them to use a calculator so that they can
solve the puzzle in 2 to 3 minutes. Answers should always be
ready for posting.
Across
Down
1. 122
1. 2(3 + 4 – 10 + 12)
5. 8[(3 + 2) + (4 + 1)]
2. (2)(2)(2)(5)
6. (9 – 2)(9 + 2)
4. (8 + 9)(9 – 8)
7. 9(2)
6. 7(10) + 8(1)
8. (20 + 3)(10 + 7)
7. (12 + 5)(12 – 5)
11. (5 + 2)(11 – 4)
9. 9(6 + 4) + 2(2)
10. (2 + 6)2
The Number Puzzle
1
2
3
5
4
3
1
2
8
9
1
1
6
7
4
10
Answer Key
1
4
8
0
1
2
3
1
4
3
1
7
1
8
7
4
9
1
6
4
9
4
34
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Main Activity:
DIVIDING POLYNOMIALS
Demonstrate the division process. Ask the students to observe the
long division process for real numbers. Let them compare long division
of polynomials with the division process for real numbers.
a) Divide 4325 by 20
Solution:
216 r. 5
4325
40
32
20
125
120
5
20
b) Divide x2 – 2x + 1 by x – 1
Solution: x - 1
x 1
x  2x  1
2
x2  x
 x 1
 x 1
0
A remainder of 0 indicates that the divisor is a factor of
the dividend. To check the division, remember the
Division Algorithm:
Dividend
= (Divisor)(Quotient) + Remainder
x2 – 2x + 1
= (x – 1)
35
(x – 1)
+
0
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
c) Divide x4 + 3x + 2 by x2 + 1
Solution:
x2 + 1
x2 1
x 4  0x 3  0x 2  3x  2
 x2
x4
-x2 + 3x + 2
-x2
-1
3x + 3
Dividend
= (Divisor)(Quotient) + Remainder
x4 + 3x + 2 =
(x2 + 1)(x2 – 1)
+ (3x + 3)
Guide Questions
In example a, how many terms are given in the dividend? In the
divisor? How about in example b? In example c?
a) What did you notice about the arrangement of terms in the dividend
in the second and third examples? Why is it necessary to arrange
the terms in descending order?
b) How is the first term of the quotient obtained? What would be the
next step?
c) What operations are involved in order to have the next dividend?
Divide the class into 5 groups that will work on a challenging problem.
Distribute the Activity Sheets. Assign one member of the group to
report the results of its work.
Extension Ideas
1. Ask the students if they know some applications of dividing
polynomials in the real world.
2. Let them work on a sample problem related to vaccination.
Yu Kamin is a genetic engineer working on a vaccine for influenza.
The number of people in a small town who catch influenza during
an epidemic is estimated to be:
36
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
N=
170t 2
t2  1
where: N represents the number of people
t represents the number of weeks from the
start of the epidemic.
One week after an epidemic starts, 170(1)2/[12 + 1] or 85 people
would have influenza. What if the epidemic lasts for a long time?
What will happen to the total population of the small town?
Complete the table below and show all the necessary
computations. What happens to the value of N as t increases?
Total Number of
Weeks
Number of People
Affected
1.5 weeks
3 weeks
7.5 weeks
12 weeks
18 weeks
21 weeks
27 weeks
Closing Activity
1. Emphasize the key learning points.
 The long division process used for real numbers can also be
used in dividing polynomials. A remainder of 0 indicates that
the divisor is a factor of the dividend.
 The division algorithm states that:
Dividend = (Divisor)(Quotient) + Remainder
2. Ask the students to arrange the basic steps in dividing a polynomial
by another polynomial from 1 to 7.
 Write the divisor (x – c) on the left-hand side of the dividend
 Divide the first term of the dividend by the first term of the
divisor to obtain the first term of the quotient.
37
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
 Write the given expression in descending order of the
exponents. Consider 0 the coefficient of any missing power of x.
 Subtract the product from the dividend to obtain the new set of
terms in the dividend. Repeat this procedure until you reach the
last term of the quotient.
 Check your answer by multiplying the quotient and the divisor.
The result should be the same as the dividend.
 Multiply the quotient by the two terms in the divisor.
Answer Key
1. Write the given expression in descending order of the
exponents. Consider 0 the coefficient of any missing power of x.
2. Write the divisor (x – c) on the left-hand side of the dividend
3. Divide the first term of the dividend by the first term of the
divisor to obtain the first term of the quotient.
4. Multiply the quotient by the two terms in the divisor.
5. Subtract the product from the dividend to obtain the new set of
terms in the dividend. Repeat this procedure until you reach the
last term of the quotient.
6. Check your answer by multiplying the quotient and the divisor.
The result should be the same as the dividend.
ASSESSMENT
Divide using long division.
1. (12x2 + 8x – 15) by (2x + 3)
2. (6x3 + x2 – 4x + 1) by (3x – 1)
3. (2y3 + 5y2 – y – 1) by (2y – 1)
4. (2m3 - 5m2 + 3) by (m2 – 2m – 4)
5. (3x2 – 2x – 4) by (x – 3)
38
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
HOMEWORK
Answer the following problems:
1. Divide 9b2 + 9b – 10 by 3b – 2
2. Simplify (5m2 – 34m – 7)(m – 7)
3. Show that 4 – n is a factor of n3 – 6n2 + 13n – 20
4. Simplify the formula for the number of people who catch influenza
170t 2
during an epidemic, n = 2
t 1
Solution
t2
170
+ 1 170t  0t  0
2
170t 2  170
170
REFERENCES
Cruz. Expanding Mathematics.
Dalton, Leroy C. Using Algebra. Third Edition.
Travers. Algebra 2 With Trigonometry.
Yu-hico. Experiencing Mathematics IV for High Schools.
39
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Student Activity 6
DIVIDING POLYNOMIALS
Challenge Problem
The area of a rectangle is represented by 6x2 + 38x + 56. Its width is
represented by 2x + 8. Point B is the midpoint of AC. ABFG is a square. Find
the length of rectangle ACED and the area of square ABFG.
Solution
Divide 6x2 + 38x + 56 by 2x + 8 to find the length of the rectangle.
3x  7
2x + 8 6 x  38 x  56
2
6 x 2  24 x
14 x  56
14x  56
0
The length of rectangle ACED is 3x + 7. The length of one side of square ABFG
is 2x + 8/2 or x + 4. The area of the square is (x + 4)2 or x2 + 8x + 16
The Figure
A
B
G
D
F
2x + 8
C
E
40
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Lesson
7
SYNTHETIC DIVISION
TIME
One session
SETTING
Math room
OBJECTIVE

At the end of this lesson, the students
should be able to find the quotient and the remainder when a
polynomial in x is divided by x – c, using synthetic division.
PREREQUISITE
Students are expected to have learned the following concepts:
 Division of polynomials using long division.
 Division of powers.
“Let x be any nonzero real number, and let a and b be any positive
integers.”
If a = b, then xa/xb = 1
If a > b, then xa/xb = xa-b
If a < b, then xa/xb = 1/xb - a
RESOURCES
 drill board or flash cards
 chart
PROCEDURE
Opening Activity
A. To check the students’ understanding of the previous lesson, give
some practice exercises on division of polynomials.
41
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Simplify each of the following:
1. 24x7/12x3
2. 12x4y2/4x2y5
3. –42x5y4/6xy3
4. –4(x – y)/20(y – x)4
B. Review the students’ assignment and ask them to solve a new set
of exercises on division of polynomials using the long method.
1. (12x2 + 8x – 15) by (x + 3)
2. (x2 + 8x + 16 ) by (x + 4)
3. (x3 + x2 – x – 1) by (x – 1)
4. (2x4 + 5x3 + 11x + 6) by (x + 3)
C. Ask the students to write their solutions on the board using long
division. Process the activity by asking these questions:
 What is the value of learning the long division method for
polynomials?
 Would you like to know a shorter way of dividing polynomials?
Main Activity
Present an illustrative example to the class:
Problem: What are the quotient and the remainder when (3x3 + x2 – 8x
– 4) is divided by (x – 2)?
Solution
3x 2  7x  6 r8
X – 2 3x3  x 2  8x  4
3x3  6x 2
7x 2  8x
7x 2  14 x
6x – 4
6 x  12
8
42
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
To check, multiply the quotient by the divisor and add the remainder.
(x – 2)(3x2 + 7x + 6) = 3x3 + 7x2 + 6x
-6x2 – 14x – 12
3
3x + x2 – 8x – 12
(3x3 + x2 – 8x – 12) + 8 = = 3x3 + x2 - 8x – 4
we write:
3x 3  x 2 - 8x - 4
8
 (3x 3  7x  6) 
x-2
x-2
Remainder is 8
Divisor is x - 2
Emphasize the fact that since we are dealing with rational
expressions, we cannot simply add the remainder, as we do with real
numbers. We have to be careful with our notation.
Discussion Ideas
A. Discuss with the students the long division method. Try to explain
how to eliminate all variables and write the problem in a more
compact form. Let the students compare the long division method
used above with the process below. Explain this in detail.
2
3
6
1
14
-8
12
-4
3
7
6
8
or
3x2 + 7x + 6
r8
Show why x – 2 is written as 2 in synthetic division. This is because
x – 2 means x – (2) or x = 2.
B. Present new sets of exercises and ask the students to solve them
using synthetic division.
1. Divide (2x3 – 9x2 + 13x – 12) by (x – 3)
Solution:
3
2
-9
6
13
-9
-12
12
2
-3
4
0
or 2x2 – 3x + 4
r0
Note: Emphasize that they should always place zero for all the
missing terms of the polynomial.
43
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
2. Divide (x5 – 4x3 + 5x2 – 5) by (x + 1)
Solution:
-1 
0
1
1
0
-1
-4
1
5
3
-8
8
-1
-3
8
-8
3
or: x4 – x3 – 3x2 + 8x – 8
5
r3
3. Find the quotient and the remainder using synthetic division.
6x3 + x2 - 4x + 1 by 3x – 1
Solution
6x3  x 2  4x  1
3x  1
1

Note 3x – 1 = 3  x  
3

This means that we can write the above expression as:
6x  x  4x  1

(3x  1)
3
1
3
Thus,
2
2x 3 
1 2 4
1
x  x
3
3
3
1
x
3
2
1
3
2
3

4
3
1
3
1
3
1

3
2
1
-1
0
(divide the expression by 3 so
we can use synthetic division)
6x3  x 2  4x  1
 (2x 2  x  1)(3x  1)
3x  1
44
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Check:
(2x2 + x – 1)(3x – 1) = 6x3 + 3x2 – 3x
-2x2 – x + 1
6x3 + x2 – 4x + 1
Guide Questions
1. Which part of the binomial divisor serves as the divisor in this
method?
2. What is done with the divisor and the numerical coefficient of the
term with the highest degree?
3. Where is the result written?
4. What operations are involved in getting the next numbers?
5. How is the last number obtained? What does a remainder of zero
mean?
6. What is the degree of the polynomial quotient?
7. Are there any restrictions in the use of synthetic division? When
can we use this method and when is this NOT applicable?
Extension Ideas
Get the meaning of the word “synthetic” from the dictionary. In your
own words, write a paragraph explaining why the process discussed
earlier is called the “synthetic division process.”
Closing Activity
a) Explain the key learning points in using synthetic division as an
alternative method for dividing polynomials.
 Synthetic division is a division process for polynomials in one
variable when the divisor is of the form x – c, where c is any real
number.
 Synthetic division is useful in the following cases:
a) finding the quotient and the remainder when a polynomial in
x is divided by x – c; and
b) determining if a binomial of the form x – c is a factor of the
given polynomial in x.
45
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
b) Recalling the steps used in dividing polynomials by synthetic
division, ask the students to complete the following:
Steps in Synthetic Division:
1. Write the numerical coefficient of the dividend in
___________order of the exponents with 0 as the coefficient of
any missing power of x.
2. Write the___________ of the divisor x – c at the left hand
corner of the coefficients.
3. Bring down the ___________of the dividend, multiply it by c and
add the result to the second column.
4. Multiply the ______obtained in the previous step by c and add
the result to the third column. Repeat this process until you
reach the last column of the dividend.
5. The third row of numbers represents the coefficient of the terms
in the quotient. The degree of the variable is one less than that
of the dividend and the rightmost number is the ________.
Answer Key
1. descending
2. constant term c
3. leading coefficient
4. sum
5. remainder
ASSESSMENT
Find the quotient and the remainder using synthetic division.
a) x3 – 2x2 + 4x + 1 by x – 2
b) 4x3 - 6x2 + 2x + 1 by x – ½
c) 2x4 + 5x3 + 11x + 6 by x + 3
46
Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
HOMEWORK
Find the quotient and the remainder using synthetic division.
1. x4 + 8x2 – 5x3 – 2 + 15 x by x – 3
2. x5 – 32 by x + 2
3.
2x3
– 7x2 – 8x + 16 by x – 4
4. 3y3 + 2y2 – 32y + 2 by y – 3
5. 76x3 – 19x2 + x + 6 by x – 3
REFERENCES
Foster and Gordon. Algebra 2 With Trigonometry—Applications and
Connections
Jose-Dilao. Advanced Algebra, Trigonometry and Statistics. Functional
Approach. 72–73.
Yu-hico. Experiencing Mathematics 4.
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Lesson
8
THE REMAINDER THEOREM
TIME
One session
SETTING
Math room
OBJECTIVES
At the end of the lesson, the students should be able to:
 state the Remainder Theorem; and
 use synthetic division and the Remainder Theorem to find the
remainder when a polynomial in x is divided by a binomial of the
form x – c.
PREREQUISITE
Students should have already learned and applied the steps used in
dividing polynomial expressions by means of synthetic and long
division.
RESOURCES
 Manila paper
 marker pen
PROCEDURE
Opening Activity: GUESS-A-REMAINDER
a) Have a drill to check on the students’ basic knowledge on division of
whole numbers.
b) Give exercises on how to use synthetic division to find the quotient
and the remainder.
1. (2x3 + 5x2 - 4x – 3) by (x + 2)
2. (2a4 + 5a3 + 7a2 – 4a + 6) by (a + 3)
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
c) Start the new lesson with a simple game following the procedure
below:
1. Ask the students to give a 3-digit number.
2. Tell them to divide the number by 9.
Guide Questions
 Can you find the remainder without dividing? How?
 When will a remainder be equal to zero?
Main Activity:
REMAINDER THEOREM
1. Develop the concept of the Remainder Theorem by presenting
sample equations. Write and solve these expressions on the board.
a. 48/7 = (6 + r6)  48 = 7(6) + r6
b. 17/5 = (3 + r2)
 17 = 5(3) + r2
Ask the students:
a. Can you write these equations another way?
48 = 7(6) + r6
17 = 5(3) + r2
b. In the new equation, what name is given to each term?
Dividend = (Divisor)(Quotient) + Remainder
2. Allow the students to solve for the quotient and the remainder using
synthetic division or long division first. Then, ask them to compare
their answers with P(1) in #2.1 and P(-1) in # 2.2
2.1
3x + 5
x–1
P(x) = 3x + 5
(1) = 3(1) + 5
=8
Therefore, the remainder is 8.
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
2x2 + 4x + 5
x+1
P(x) = 2x2 + 4x + 5
P(-1) = 2(-1)2 + 4(-1) + 5
=2+1
=3
Therefore, the remainder is 3.
2.2
3. Find the remainder using the Remainder Theorem:
(9a3 + 3a + 6a2 + 9) divided by (3a – 3)
Solution
9a3  3a  6a2  9
(find the remainder)
3a  3
Can be written as:
9a3  3a  6a2  9 3a3  a  2a2  3
=
(divide the expression
a 1
3(a  1)
by 3 so that the divisor is of
the form a – c)
By the Remainder Theorem:
3(1)3 + 1 +2(1)2 + 3 = 3 +1 + 2 + 3 = 9
Multiply this by 3, since 3a – 3 = 3(a – 1)
 The remainder when 9a3 + 3a + 6a2 + 9 is divided by
3a – 3 is 9(3) = 27
Check:
3a2 + 5a + 6
3a  3 9a  6a2  3a  9
- 9a3 – 9a2 _______
15a2 + 3a
-15a2 + 15a___
18a + 9
-18a +18
27
3
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
4. Divide the students into small groups. Have them work on the given
set of problems provided in the attached Student Activity Sheet.
Discussion Ideas
1. What is the relation between the remainder and the value of the
polynomial at x = c when the polynomial in x is divided by a
binomial of the form x – c?
2. How is the remainder obtained when the polynomial in x is divided
by a binomial of the form x –c?
3. What happens if the remainder is zero?
Note: The value of the polynomial at x = c is equal to the remainder
when the divisor is x – c.
Closing Activity
a) Emphasize the key learning points:
 Division Algorithm for Polynomials
For each polynomial P(x) of a positive degree, and for any
number c, there exist unique polynomials Q(x) and R(x) such
that:
P(x) =[ (x – c) Q(x)] + R(x),
Where: Q(x) is of degree n – 1
R(x) is the remainder
 Remainder Theorem
If a polynomial P(x) is divided by x – c, then the remainder is
P(c).
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Proof of the Remainder Theorem
1. P(x) = (x – c)  Q(x) + R(x)
Division Algorithm for Polynomials
2. P(x) = (x – c)  Q(x) + R
Definition of Division of Polynomials
R(x) must have a degree less than the
degree of (x – c).
`
Thus, R(x) = R is a constant (which may or
may not be 0)
3. P(c) = [(c – c)  Q( c )] + R
Equation (2) is true for all x. Therefore x = c
4. P(c) = 0 x Q ( c ) + R
Multiplication Property
5. P(c) = R
Identity
Hence, the remainder R is equal to P(c).
b) Ask the students to relate synthetic division with the Remainder
Theorem in terms of evaluating polynomials.
Answer Key
 Synthetic division can be used as a convenient way to find the
values of polynomial functions P(c). Hand in hand with this
process is the Remainder Theorem. The remainder r obtained
in synthetic division is indeed equal to P(c).
ASSESSMENT
Find the remainder using the Remainder Theorem.
1. (2y3 – 5y2 – 8y – 50) divided by (y – 5)
2. (3y3 + 2y2 – y + 5) divided by (y + 2)
3. (x3 + 4x – 7) divided by (x – 3)
4. (8x4 + 2x + 4x3 + 1) divided by (x + ½)
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
HOMEWORK
Using the Remainder Theorem, find the remainder of the following:
1. (12x4 + 11x – x2 – 40x3 –90) by (x – 3)
2. (2x4 – 4x3 + 9x2 + 2x – 5) by (x – 1)
3. (4y3 – 6y2 + 2y + 1) by (2y – 1)
4. (y4 + 8y2 – 5y3 – 2 + 15y) by (y – 3)
5. (3m3 – 2m2 + 2m – 3) by (m + 3)
REFERENCES
Foster, Gordon. Algebra 2 With Trigonometry—Applications and
Connections.
Jose-Dilao. Advanced Algebra, Trigonometry and Statistics. Functional
Approach. 74–77.
Yu-hico. Experiencing Mathematics 4.
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Student Activity 8
REMAINDER THEOREM APPLICATIONS
Solve the following polynomial functions:
A.
Use the Remainder Theorem to find the value of x4 – 4x2 + 12x – 9
when x =2.
B.
Use the Remainder Theorem to find the value of x4 – 4x3 + 12x – 9
when x = 1.
C.
Determine whether x – 3 is a factor of 2x3 – 3x2 – 12x + 9.
If x – 3 is a factor, name the other factor of the polynomial.
D.
Which of the following binomials are factors of x4 – 3x2 + 6x - 4?
a.)
x+2
b.)
x–1
c.)
x+3
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Lesson
9
THE VALUES OF POLYNOMIAL FUNCTIONS
TIME
One session
SETTING
This lesson must be done inside the math room. This is simply a
continuation of the past two lessons on synthetic division and the
Remainder Theorem. If the class learned the past lessons quickly, this
lesson may be integrated into those lessons as no new concepts are
presented here.
OBJECTIVES
At the end of the lesson, the students should be able to further develop
their skills in determining the:
 quotient of a polynomial divided by a binomial of the form (x - c)
using synthetic division; and
 value of a polynomial using synthetic division and the Remainder
Theorem.
PREREQUISITE
Students should have already learned the Remainder Theorem and
synthetic division.
RESOURCES
 Manila paper
 marker pen
PROCEDURE
Opening Activity
A. Check the students’ assignments and review the class on
evaluating polynomial functions by means of a short contest:
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
B. Form two teams to evaluate polynomials for integral values of the
variable. One team will use direct substitution while the other team
will use synthetic division. Ask the students to work on the following
practice set.
Task: Determine the remainder in each of the following polynomial
expressions using synthetic division.
a.) y3 – 4y2 – 2y + 5; y – 1
b.) 2x3 – 3x2 + 2x – 8; x – 2
c.) 4m3 + 4m2 + m + 3; m – 4
Main Activity
A. Give the following set of questions for students to solve as a group.
The outputs of each group will be presented to the class.
1. Given the polynomial 8x3 + 6x + 7, find its degree and leading
coefficient.
(Answer: Since 3 is the highest exponent, the polynomial is of
degree 3 and 8 is its leading coefficient.)
2. To find the quotient and the remainder when 3x3 – 5x + 10 is
divided by x – 2, what are the initial steps you must take?
(Answer: Arrange the terms in order of descending power and
write zero for the missing terms.)
3. Using long division and synthetic division, find the quotient and
remainder. Compare your answers.
Long Division
3x 2  6x  7
3x3  6x 2
x–2
3x3  6x 2
6x 2  5x
6x 2  12x
7x  10
7x  14
24
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Synthetic Division
2 3
3
0
-5
10
6
12
14
6
7
24
3x2 + 6x + 7
24 is the remainder
3x2 + 6x + 7
is the quotient
 The quotient is 3x2 + 6x + 7 and the remainder is 24.
Using either of the methods, students should be able to
find the same quotient and the remainder.
5. Using the Remainder Theorem, find the value of the polynomial
at x = 2.
f(x) = 3x3 – 5x + 10
f(2) = 3(2)3 – 5(2) + 10
= 3(8) – 10 + 10
f(2) = 24
By the Remainder Theorem, the remainder will be equal to the
value of the polynomial at x = 2.
B. Present another example:
Divide (2x3 – 4x2 – 5x + 5) by (x + 1)
Since synthetic division is much shorter, you may use this method
to save time.
-1
2
2
-4
-5
5
-2
6
-1
-6
1
4
2x2 – 6x + 1 r 4
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
f(x) = 2x3 – 4x2 – 5x + 5
f(-1) = 2(-1)3 – 4(1)2 – 5(-1) + 5
= 2(-1) – 4(1) + 5 + 5
= –2 –4 + 5 + 5
= –6 + 10
f(-1) = 4
Discussion Ideas
1. When we divided 3x3 – 5x + 10 by x – 2, what did we use as divisor
in the synthetic division?
2. What can you say about the remainder and the value of the
polynomial?
Closing Activity
1. Synthesize the key points of the lesson by saying:
If a polynomial f(x) is divided by x – r until a remainder without x is
obtained, this remainder is equal to f (r).
2. Ask students: How do you find the value of a polynomial?
Answer: Apply the Remainder Theorem and synthetic division.
ASSESSMENT
Find the value of the polynomials using the Remainder Theorem and
synthetic division.
1. (2y3 – 5y2 – 8y – 50) divided by (y – 5)
2. (3y3 + 2y2 – y + 5) divided by (y + 2)
3. (x3 + 4x – 7) divided by (x – 3)
4. (9a3 + 3a + 6a2 + 9) divided by (3a – 3)
5. (8x4 + 2x + 4x3 + 1) divided by (x + ½)
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
HOMEWORK
Using the Remainder Theorem, find the value of the polynomials.
1. (12x4 + 11x – x2 – 40x3 –90) by (x – 3)
2. (2x4 – 4x3 + 9x2 + 2x – 5) by (x – 1)
3. (4y3 – 6y2 + 2y + 1) by (2y – 1)
4. (y4 + 8y2 – 5y3 – 2 + 15y) by (y – 3)
5. (3m3 – 2m2 + 2m – 3) by (m + 3)
REFERENCES
Foster, Gordon. Algebra 2 With Trigonometry–Applications and
Connections
Jose-Dilao. Advanced Algebra, Trigonometry and Statistics. Functional
Approach. 74–77.
Yu-hico. Experiencing Mathematics 4
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Lesson
10
THE FACTOR THEOREM
TIME
One session
SETTING
Math room
OBJECTIVE
At the end of this lesson, the students should be able to use the Factor
Theorem and other concepts to find the zeroes of the polynomial
function of degree greater than two.
PREREQUISITE
Students should have learned the zeroes of polynomial functions.
RESOURCES
 Chart
PROCEDURE
Opening Activity
A. Have students practice their skills in factoring by solving the
following:

4y – 24

15a3 + 20a

4x2 –81

y2 + 18y+ 81
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
B. Conduct a review on finding the zeroes of a polynomial function.
Present these examples as written on Manila paper for the
students’ verification.
1. Given f(x) = 4x + 2, find the zero of the function.
4x + 2 = 0
4x = 2
x = -2/4 or –1/2
2. If f(x) = x2 + 3x – 10, find the zeroes of the function:
x2 + 3x – 10 = 0
(x + 5)(x – 2) = 0
x = –5; x = 2
The zeroes of the function are –5 and 2.
Main Activity
A. Begin the new lesson by saying:
A linear function has only one zero, while a quadratic function,
has at most two zeroes. If the degree of a given polynomial
function is n, how many zeroes does the function have? How
are the zeroes of a function found?
a. What is/are the zero/zeroes of the function in the previous
examples?
b. How many zeroes are there in each function?
B. Develop the students’ understanding of the Factor Theorem by
asking them to analyze the example below (already prepared in
Manila paper for the presentation).
a) P(x) = x3 – x2 – 10x – 8
b) P(x) = x3 + x2 – x – 1
C. Discuss the Factor Theorem and relate this to the Remainder
Theorem. Explain how these two theorems complement each other.
Also discuss how they can be used to solve problems involving
roots and the factorization of polynomials with degree higher than
2.
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
The Remainder Theorem states that:
“When the polynomial P(x) is divided by x-c, the Remainder is
P(c)”
The Factor Theorem
“Given the polynomial P(x). If P(c) = 0, then x-c is a factor of
P(x). Conversely, if x-c is a factor of P(x), then P(x) = 0.
If the remainder when P(x) is divided by (x-c) is zero, then (x-c)
is a factor of P(x).
PROOF
Suppose P(c) = 0. By the Remainder Theorem, when P(x) is
divided by (x–c), the remainder R = P(c) = 0
Then, P(x) = (x-c)  Q (x) + R becomes
P(x) = (x–c)  Q (x) + 0
P(x) = (x–c)  Q (x)
Therefore (x–c) is a factor of P(x)
D. Present some illustrative examples depicting the Factor Theorem
1) Ask the students to guess the value of x in the given function:
P(x) = x3 – x2 – 10x – 8
Substitute 4, -2, -1 to the given function.
If x = 4,
x3 – x2 – 10x – 8
43 – 42–10(4) – 8 = 0
(x – 4) is a factor of P(x).
If x = -2,
X3 – x2 –10x – 8
(-2)3 – (-2)2 – 10(-2) – 8 = 0
(x + 2) is a factor of P(x)
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
If x = -1
x3 – x2 – 10x – 8
(-1)3 – (-1)2 – 10(-1) – 8 = 0
(x + 1) is also a factor of P(x).
Therefore, the zeroes are 4, –2, and –1.
2) P(x) = x3 + x2 – x – 1
Substitute 1, 2, -1 to the given function.
If x = 1
x3 – x2 – x – 1
13 + (1)2 – 1 – 1
1+1–1–1=0
(x–1) is a factor of P(x)
If x = 2
x3 + x2 – x – 1
23 + 2 2 – 2 – 1
8+4–2–1
12 – 2 – 1 = 9
(x – 2) is not a factor of P(x) and 2 is not a zero of
P(x)
If ax = -1
x3 + x2 – x – 1
(-1)3 + (-1)2 – (-1) – 1
-1 + 1 + 1 –1 = 0
(x + 1) is a factor of P(x).
 Therefore, the only zeroes of the function are 1 and –1.
 f(x)=x3 + x2 – x – 1 is a polynomial of degree 3.
 We found that 1 and –1 are roots of this equation.
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
 Using synthetic division, we have:
1
1
1
1
-1
2
-1
1
-1
1
2
-1
1
-1
0
-1
1
1
-1
0
2
0
p(x) = x3 + x2 – x - 1
p( x)
 x 2  2x  1
x 1
p( x )
 x 1
( x  1)( x  1)
p( x )
1
( x  1)( x  1)( x  1)
Note: From this, we can see that p(x) can be factored as:
 p(x) = x3 + x2 – x – 1 = (x – 1)(x +1)(x + 1)
 because –1 served as a root twice, we say –1 is a root of
MULTIPLICITY 2
E. Divide the class into small groups. Have the groups work on the
following sets of polynomial functions:
 Which of the following binomials, (x – 1), (x – 2), (x + 2), (x –3),
(x + 3) are divisors of the following expressions? Find the
zeroes of their related functions.
a. x3 + 2x2 – 9x – 8
b. x4 – 4x3 – x2 + 16x – 12
c. x4 – 2x3 – 7x2 + 8x + 12
Discussion Ideas
1. What are the degrees of the functions in numbers 1 and 2?
2. How many zeroes are there in numbers 1and 2?
3. How are 4, -1, -2 related to the constant term given in example
number 1? How are 1, 2, -1 related to the constant term in example
number 2?
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
4. Why is there a need to substitute these possible factors to the
function?
5. How are the zeroes of a polynomial function related to the constant
term of the polynomial?
6. When is a certain number a zero of a polynomial function? When is
it not a zero?
7. What are the steps in finding the zeroes of P(x) using the Factor
Theorem?
Closing Activity
Ask students to determine the steps in finding the zeroes of a
polynomial P(x). Consolidate their responses to come up with the
following steps:
1. Substitute possible values of c for x in the given polynomial P(x).
2. Use the Remainder Theorem. If P(c) = 0, then (x – c) is a factor of
P(x) by virtue of the Factor Theorem.
3. Equate each factor to zero to get the zeroes of the function.
4. The number of zeroes of a function is less than or equal to the
degree of the given function.
ASSESSMENT
Find the zeroes of the function:
1. P(x) = x3 + 3x2 – 2x – 6
2. P(x) = x3 – x2 – 4x + 4
3. P(x) = 3x3 – x2 + 12 – 4
HOMEWORK
Find the zeroes of the following polynomials using the Factor Theorem.
1.) P(x) = x3 – 12x2 + 41x - 4
2.) P(x) = x4 – 3x2 + 2
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
REFERENCES
Foster, Gordon. Algebra 2 With Trigonometry–Applications and
Connections
Jose-Dilao. Advanced Algebra, Trigonometry and Statistics. Functional
Approach. 82–84.
Mathematics IV. Advanced Algebra, Trigonometry and Statistics. 2002
BEC.
Yu-hico. Experiencing Mathematics 4.
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Lesson
11
ZEROES OF POLYNOMIAL FUNCTIONS: A RECALL
TIME
One session
SETTING
Math room
OBJECTIVES
At the end of this lesson, the students should be able to:
 apply the Factor Theorem, factoring techniques, synthetic and long
division and the Rational Root Theorem; and
 find the zeroes of polynomial functions of degree greater than 2.
PREREQUISITE
The students are expected to have learned the following concepts:
 Finding the zeroes of linear and quadratic functions, synthetic and
long division as well as the Remainder Theorem to solve for the
quotient and the remainder of a polynomial function
RESOURCES
 graph board
 chart
 colored chalk
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
PROCEDURE
Opening Activity
A. Check the students’ assignment and review finding the quotient and
the remainder, if necessary. Ask the students to find the quotient
and the remainder of the following polynomials:
a.) (2y3 – 5y2 – 8y – 50) divided by (y – 5)
b.) (3y3 + 2y2 – 7) divided by (y – 3)
B. Introduce the lesson by reviewing how to find the zeroes of a
polynomial function.
 Recall the meaning of “zeroes of a polynomial function”. Ask the
students for illustrations of quadratic and linear functions.
 Ask them to extend the meaning of zeroes of polynomials of
degree > 2 by illustrating this graphically.
 Present the graphs of the following linear and quadratic
functions, and ask the students to answer the guide questions
that follow.
Sample Graphs of Linear Functions
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Graphs of Quadratic Functions
y=x2
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Guide Questions
For each function:
 At what point/points does the graph cross the x-axis?
 Notice that in each of the points of intersection with the y-axis the
value of f(x) is zero. What is the value of x when f(x) = 0?
 What are these x-values called?
1. Ask the students to find the value of x when f(x) = 0
a. Find the zero of f(x) = 3x – 5.
F(x) = 3x – 5
0 = 3x - 5
3x = 5
x = 5/3
b. Find the zeroes of f(x) = x2 – 5x + 4
F(x) = x2 – 5x + 4
x2 – 5x + 4 = 0
(x – 4)(x – 1) = 0
x = 4, x = 1
 The zeroes of the function are 4 and 1
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
2. Illustrate the procedure for finding the zeroes of a polynomial
function by factoring. Let the students find the zeroes of the
following functions:
a) P(x) = (x – 5)2 (x + 3)
double root
Zeroes are 5 and –3, 5 being a
b) P(x) = 3x3 + 9x2 – 30x
Zeroes are 0, -5 and 2.
3. Recall the Factor Theorem, which states that:
“Given the polynomial f(x). If the remainder when f(x) is divided
by (x-c) is zero, then (x-c) is a factor of f(x).
Main Activity
Part I
Let individual students try out some integers in the synthetic division
for each polynomial and find the zeroes of the given P(x):
a.) P(x) = x3 + 9x2 + 23 x + 15
b.) P(x) = x4 + 5x3 + 5x2 – 5x – 6
Answers
a.) Zeroes are –1, -3, and –5
b.) Zeroes are 1, -1, -3, and –2.
Guide Questions
 For P(x) = x3 + 9x2 + 23x + 15, what numbers may possibly be
eliminated? (0 and positive values). Why?
 How many negative integers should one try out as possible
zeroes?
 When should one stop dividing synthetically?
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Discussion Ideas
Based on the previous lesson, the students were able to find the
factors of polynomials of higher degree. At this point, the students
shall find the zeroes of such polynomials.
 The polynomial f(x) = 4x3 – 4x2 – 8x is of the third degree;
therefore, we expect the function to have three zeroes.
 Look for a common monomial factor among the terms of the
function 4x3 – 4x2 – 8x and factor this out: 4x (x2 - x – 2).
 Note that x2 – x – 2 can be factored further as (x – 2)(x + 1).
Therefore, 4x3 – 4x2 – 8x = 4x (x – 2)(x + 1)
 Equating all the factors to zero since f(x) = 0 we find that if:
4x = 0
Then:
x=0
x–2=0
x+1=0
x=2
x = -1
Note: This method is applicable if the given polynomial can
easily be factored.
Part II. Present another example for the students.
f(x) = x4 – x3 – 7x2 + x + 6
The factors of 6 are: 2 and 3, -2 and -3, 6 and 1, -6 and -1?
Solution
Because f(x) is not readily factorable, we make use of a theorem
called the RATIONAL ROOT THEOREM:
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Rational Root Theorem
Given a polynomial p(x0 = anxn+an-1xn-1 + … + a0. Then the rational
b
roots of the polynomial are of the form o , where bo and bn are
bn
divisors of ao and an, respectively.
We apply this theorem to our examples:
 fx) = x4 – x3 – 7x2 + x + 6 where, an = 1 and a0 = 6.
 The divisors of ao = 6 are: 1, 2, 3, 6, -1, -2, -3, -6
 The divisors of ao = 1 are: 1, -1
The only possible rational roots of f(x) are:  1,  2,  3,  6
We choose which of these numbers to use in synthetic division:
Note
1. Clearly, 1 is a root, since f(1) = 1 – 1 – 7 + 1 + 6 = 0. Thus,
the remainder when f(x) is divided by (x-1) is zero. By the
FACTOR THEOREM, (x – 1) is a factor of f(x).
2. Similarly, -1 is a root since f(-1) = 1 + 1 – 7 – 1 + 6 = 0
Because 1 is a root, we look for the quotient
f (x)
by Synthetic
x 1
Division
1
1
-1
1
-7
0
1
-7
6
-6
f(x) = x4 – x3 – 7x2 + x + 6
-1
1
0
-1
-7
1
-6
6
0
f(x)
= x3 – 7x - 6
( x  1)
1
-1
-6
0
f (x)
= x2 – x – 6
( x  1)( x  1)
Note: We can continue the synthetic division in this way until all the
rational roots have been found.
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Chapter 1: Working with Polynomial Functions
So far, we found two roots of f(x): 1 and –1.
Using the Factor Theorem, we know that:
f(x) = (x – 1)(x + 1) (x2 – x – 6)
= (x – 1)(x + 1) (x –3)(x + 2)
Thus, the roots of f(x) are 1, -1, 3 and –2.
Note: After the above example, the teacher may give another
similar example for the students to work on by themselves.
Example
Find all the zeroes of f(x) = x4 + 2x3 – 3x2 – 8x – 4. Use all available
techniques.
Solution
a) By the Rational Root Theorem, an = 1, ao = –4
 possible rational roots  1,  2,  4.
b) By using the Remainder Theorem: we try to guess the roots
f(1) = 1 + 2 – 3 – 8 – 4 = 3 – 3 – 8 – 4  0  1 is not a root
Thus, x – 1 is NOT a factor of f(x))
f(-1) = 1 - 2 – 3 + 8 – 4 = 1 – 5 + 8 – 4 = 0

-1 is a root of f(x), and so (x + 1) is a factor of f(x)
c) Using Synthetic Division find the quotient:
-1
-1
1
1
1
2
-1
-3
-1
-8
4
1
-1
-4
0
-4
4
0
-4
0
-4
4
0
f(x) = x4 + 2x3 – 3x2 – 8x - 4
f ( x)
 x3  x 2  4x  4
x 1
(we use the remaining possible
numbersto check if they are roots)
f ( x)
 x2  4
( x  1)( x  1)
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Chapter 1: Working with Polynomial Functions
Note: It is possible that a number is a root more than once. In this
case, -1 is of MULTIPLICITY 2 because it is a root 2 times
So far, we know by Synthetic Division and the Factor Theorem
that:
f(x) = (x + 1)(x + 1)(x2 – 4)
= (x + 1)(x + 1)(x + 2)(x – 2)
Hence, the roots of f(x) = x4 + 2x3 – 3x2 – 8x – 4 are –1, –2 and 2,
where –1 is of multiplicity 2.
Part III. Divide the students into small groups. Ask them to find the
zeroes of each of the given functions:
1. f( x ) = 2x3 – x2 – 13x – 6
2. f( x ) = 4x3 + 13x2 – 37x – 10
3. f( x ) = 6x3 + 25x2 + 2x – 8
4. f (x) = 6x3 + 19x2 + 11x - 6
Guide Questions
 What are the ways of determining the zeroes of a function?
 Without graphing, how are the zeroes of a function determined?
 How many zeroes are there in a:

linear function?

quadratic function?

polynomial of degree n?
Closing Activity
Ask the students: What is meant by “zeroes of a function”?
1. A zero of a polynomial function is the value of the variable x, which
makes the polynomial equal to zero, or f(x) = 0.
2. The zeroes of P(x) are the roots of the equation P(x) = 0. A linear
function has only one zero, while a quadratic function has at most
two zeroes.
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3. The zeroes of quadratic function can be found by factoring,
completing the square, or by using the quadratic formula.
4. In general, a polynomial of degree n will have at most n real roots
or zeroes.
ASSESSMENT
A. Factor the following polynomials into linear factors:
1. f(x) =x4 + 3x3 – 3x2 – 7x + 6
2. f(x) =x4 + x3 – 11x2 – 9x + 18
B. Find the zeroes of the following:
a. f(x) = x + 3
b. f(x) = x2 – x – 30
c. f(x) = x2 – 2
d. f(x) = x2 + 4x + 4
HOMEWORK
Graph the following functions:
1. f(x) = 2x – 7
2. f(x) = 3x + 8
3. f(x) = x2 – 8x – 48
4. f(x) = x2 + 9x - 22
REFERENCES
Foster, Gordon. Algebra 2 With Trigonometry–Applications and
Connection
Jose-Dilao. Advanced Algebra, Trigonometry and Statistics. Functional
Approach. 78–87.
Yu-hico. Experiencing Mathematics 4.
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Chapter 1: Working with Polynomial Functions
Lesson
12
GRAPHS OF POLYNOMIAL FUNCTIONS (Part 1)
TIME
One session
SETTING
Math room
OBJECTIVES
At the end of the lesson, the students should be able to:
 graph the polynomial functions of degree greater than two; and
 locate the zeroes of a polynomial function by estimating its position
on a graph
PREREQUISITE
The students must be familiar with the Factor Theorem, the Remainder
Theorem, the Rational Root Theorem, and with synthetic and long
division
RESOURCES
 chart
PROCEDURE
Opening Activity
a) Check the students’ basic knowledge on evaluating a polynomial
function. Ask them to find the value of P(x):
Given:
P(x) = x3 – 3x2 – x + 10
Find:
P(2), P(-2), P(3)
b) Introduce the new lesson by asking students to illustrate the graphs of
the following:
a) F(x) = a?
b) F(x) = ax + b?
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c) F(x) = ax2 + bx + c?
d) F(x) = x3?
c) Allow the students to illustrate the different cases or possibilities for
the first three. Let the students graph each of the functions.
Main Activity
Note! Prepare sample graphs for presentation to the students:
1. Point out the zeroes of the functions graphed on the board. Then,
explain the zeroes of the polynomial functions.
2. Let the students observe the graphs of the polynomial function
f(x) = xn, when:
a) n is an integer greater than zero.
b) n is even
c) n is odd
3. Give some examples:
a. Consider the function f(x) = x4 – 5x2 + 4
Using the different techniques for finding the roots of a
polynomial function, the roots of f(x) are found to be: –2, -1, 1
and 2.
The graph of f(x) looks like this:
4
-2
-1
1
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Chapter 1: Working with Polynomial Functions
Notice that the graph of f(x) intersects the x-axis at the zeroes of
the function (f(x)= x4 – 5x2 + 4) or where y is equal to zero.
Because f(x) is of degree 4, it has, at most, 4 real roots.
b. On the other hand, the graph of f(x) = x4 – 2x3 – 2x – 1 looks like
this:
-1
-1
1
 How many zeroes does f(x) have? Try to verify your answer by
using previous techniques learned.
4. Individual Activity
a) Ask the students to follow the steps in graphing the polynomial
function: f(x) = (x – 2) (x + 1)(x – 1), as described in the
textbook.
Remind them that the zeroes of a polynomial function are the
roots of the related polynomial equation.
b) Find the zeroes of the following polynomial functions, then graph.

P(x) = x3 – 9x + 1

P(x) = x3 – 4x - 2
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Discussion Ideas
1. Describe the graph of a polynomial function of degree greater than
2. Differentiate it from linear and quadratic functions.
2. What is meant by “the zeroes of a function”? How can you
determine the zeroes from the graphs?
3. What does the graph of f(x) = xn look like when n is even? What if
n is odd?
4. How would you describe the graph of f(x) = x5? What about
f(x) = x6?
5. What is the basic step in graphing any function?
Closing Activity
Ask the students:
What can you say about the graphs of polynomial functions?
 The graph of a polynomial function has the equation of the form
f(x)=x n, where n is an integer greater than zero. It is continuous
such that it has no breaks and has smooth rounded turns.
ASSESSMENT
Graph the function P(x) = x3 – 3x + 1, and estimate the real roots to the
nearest one-half unit.
HOMEWORK
Use a graph to estimate the real roots to the nearest half unit of the
equation P(x) = 0.
 P(x) = x3 – x2 – 2x – 1
 P(x) = x3 – x2 – 2x + 1
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REFERENCES
Foster and Gordon. Algebra 2 With Trigonometry–Applications and
Connections.
Jose-Dilao. Advanced Algebra, Trigonometry and Statistics. Functional
Approach. 88–92.
Math IV: Advanced Algebra, Trigonometry and Statistics–2002 BEC
Yu-hico. Experiencing Mathematics 4.
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Chapter 1: Working with Polynomial Functions
Lesson
13
GRAPHS OF POLYNOMIAL FUNCTIONS (Part 2)
TIME
One session
SETTING
This lesson can be done inside the math room. This lesson is
supplementary to the previous lesson. The teacher should focus more
on polynomial functions of degree >2 and not spend too much time
recalling quadratic functions.
OBJECTIVES
At the end of this lesson, the students should be able to:
 sketch the graph of a given polynomial function by assigning points;
 apply previously learned theorems and concepts in graphing
polynomials of degree > 2.
PREREQUISITE
The students must have enough skills in applying the Factor Theorem,
factoring techniques, synthetic division and depressed equations to
find the zeroes of polynomial functions of degree greater than two.
RESOURCES




graphing boards
graphing papers
colored chalk
chart
PROCEDURE
Opening Activity
A. As a practice exercise, ask the students: Which of the following
functions is linear? Which are quadratic?
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1. P(x) = 3x + 5
2. P(x) = x2 – 5x + 4
3. P(x) = x2 - 4
B. To review the graph of a polynomial function, show 5 samples of
graphs of linear and quadratic equations. Let the students identify
the kinds of graphs presented.
1) Present three linear functions, e.g.:
 f(x) = x + 2
 f(x) = -2x – 1
 f(x) = x – 5
Then ask the students to draw the graphs of each of these and
let them identify the following:
a. line
b. slope
c. x-intercept
d. y-intercept
Ask the students: How is the graph of a linear function drawn?
1) Present two quadratic functions, such as:
 f(x) = x2 – 3x – 4
 f(x) = -x2 + 2x + 24.
Give the students a few minutes to draw their graphs and let
them identify the following:
a) parabola
b) vertex
c) axis of symmetry
d) direction of opening
e) x-intercepts
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Ask the students: How is the graph of a quadratic function
drawn?
C Present a third-degree polynomial function in factored form, e.g.,
P(x) = (x – 2)(x +1)(x – 1). Prepare a Cartesian coordinate plane on
the board and guide the students in sketching the graph of this
function.
Main Activity
A. Compare the graphs of linear and quadratic functions:
 From previous discussions, we learned of the graphs of two
polynomial functions, the linear and the quadratic functions.
The graph of a linear function f(x) = ax + b, where a and b are
real numbers, is a straight line; the graph of a quadratic
function, f(x) = ax2 + bx + c, where a, b, and c are real
numbers and a is not zero, is a parabola
B. Allow each group to work on the following problems:
1. What do you think will happen if:
a) g(x) is decreased by 2 as in g(x)= x3 – 2?
b) h(x) is increased by 3 as in h(x) = x3 + 1
2. Sketch each of them and compare them with the graph of
f(x) = x3.
Discussion Ideas
a. Consider the following important points:
1) The zeros of a function are really the x-intercepts of the
function. These can be plotted easily.
2) The values of the function in the following intervals will provide a
picture of the behavior of the graph in these intervals.
x < -1
-1< x < 1
1<x<2
x>2
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b. Have the students prepare a table of values where 2 to 3 values for
x from each of the above intervals are used. These should be
enough for the students to have an idea of the behavior of the
graph in each interval. Let them show the table of values and the
resulting graph:
Closing Activity
a. Ask the students: How do you graph a given polynomial function?
b. Summarize their responses:
 To graph a polynomial function, first find the zeroes of the
function. Then construct a table of values of the given variables
and the corresponding value of the polynomial function. These
values should represent points in the different intervals into
which the zeroes of the function divide the x-axis. Plot these
values on a Cartesian coordinate plane. If the points are rather
far apart, assign fractional values to the variable to determine
the shape of the curve more accurately.
ASSESSMENT
1. Draw the graph of the polynomial function: f(x) = x3- 4x2 + x + 3
2. What are the roots of the equation?
3. Show the table of values.
4. What are the x-intercepts?
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Chapter 1: Working with Polynomial Functions
HOMEWORK
1. Graph the function f(x) =
1 3
x
3
2. Graph the function f(x) = x3 + 2x2 – 4x – 1
3. Graph the function f(x) = (x – 1)4
REFERENCES
Foster and Gordon. Algebra 2 With Trigonometry—Applications and
Connections
Jose-Dilao. Advanced Algebra, Trigonometry and Statistics. Functional
Approach. 88–92.
Math IV: Advanced Algebra, Trigonometry and Statistics – 2002 BEC
Yu-hico. Experiencing Mathematics 4.
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Chapter 1: Working with Polynomial Functions
Lesson
14
GRAPHS OF POLYNOMIAL FUNCTIONS (Part 3)
TIME
One session
SETTING
Math room
OBJECTIVE
At the end of this lesson, the students should be able to:
 solve polynomial functions using the most logical procedure; and
 graph polynomial functions of degree greater than two.
PREREQUISITE
Students must have enough skills in applying the Factor Theorem,
factoring techniques, synthetic division and depressed equations to
find the zeroes of polynomial functions of degree greater than two.
RESOURCES




graphing boards
graphing paper
colored chalk
chart
PROCEDURE
Opening Activity
A. Have students review plotting of points. Ask them to identify the
following features of the graph of a polynomial function:
a. vertex
b. axis of symmetry
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c. x-intercepts
d. y-intercepts
B. The teacher may choose to solve items in the assignment.
Main Activity
Students have learned to identify the zeroes and turning points of a
function from a prepared graph. With this, they can proceed to the next
activity. Present the solution and graph of a polynomial function to the
students for observation. Example: f(x) = x3 – 3x + 2
We can use the Rational Root Theorem to find out the possible roots of
f(x). In f(x) = x3 –3x+ 2, the factors of 2 are 1, -1, 2, -2. We test each of
these to determine the roots:
f(x) = x3 – 3x + 2
f(1) = 13 – 3(1) + 2
=1–3+2
= 0, then 1 is a zero
f(x) = x3 – 3x + 2
f(-1)= (-1)3 – 3(-1) + 2
= -1 + 3 + 2
= 4, then –1 is not a zero
f(x) = x3 – 3x + 2
f(2) = 23 – 3(2) + 2
=8–6+2
= 4, then 2 is not a zero
f(x) = x3 – 3x + 2
f(-2) = (-2)3 – 3(-2) + 2
= -8 + 6 + 2
= 0, then –2 is a zero
Ask the students to prepare a table of values using numbers
representing the various intervals. Then, plot the graph of the function.
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Chapter 1: Working with Polynomial Functions
Discussion Ideas
Ask the students to look at the graph and to answer the following
questions:
1) What are the zeroes of the function?
2) What happens to the graph in each interval?
3) What are the factors of f(x)?
4) Which root has a multiplicity of 2?
Extension Ideas
Ask the students to look for a problem in science that make use of the
zeroes of a polynomial function.
Closing Activity
Emphasize the key points of the lesson by asking:
1) How do you graph a given polynomial function?
2) How do you find the zeroes of a polynomial function?
ASSESSMENT
Solve and graph the function f(x) = x3 + 6x2 + x + 6
 What are the zeroes of the function?
HOMEWORK
Solve and graph:
1. f(x) = x3 – 8
2. f(x) = x4 + 3x3 + 2x2 - 2
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Chapter 1: Working with Polynomial Functions
REFERENCES
Foster and Gordon. Algebra 2 With Trigonometry—Applications and
Connections,
Jose-Dilao. Advanced Algebra, Trigonometry and Statistics. Functional
Approach. 88–92.
Math IV: Advanced Algebra, Trigonometry and Statistics – 2002 BEC.
Mathematics IV – SEDP Series.
Yu- hico. Experiencing Mathematics 4.
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Advanced Mathematics: Unit II- Advancing with Polynomials and Exponents
Chapter 1: Working with Polynomial Functions
Lesson
15
GRAPHS OF POLYNOMIAL FUNCTIONS (Part 4)
TIME
One session
SETTING
Math room
OBJECTIVES
At the end of this lesson, the students should be able to accurately
draw the graph polynomial functions of degree greater than two.
PREREQUISITE
Students must have enough skills in applying the Factor Theorem,
factoring techniques, synthetic division and depressed equations to
find the zeroes of polynomial functions of degree greater than two.
RESOURCES




Manila paper
graphing papers
pencil
ruler
PROCEDURE
Opening Activity
1. Check the students’ assignments.
2. Divide the class into 5. Let each group draw the graph of any given
polynomial function on A3–sized bond paper. Let them emphasize
the curve using a marker pen or crayon. Ask them to create an
artwork using the curve as background.
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Chapter 1: Working with Polynomial Functions
3. Ask each group to describe its graph. They may add something like:
 In graphing polynomial functions, the zeroes of the polynomial
function are the roots of the polynomial equation which
corresponds to f(x). The real roots of any polynomial equation
correspond to the point of intersection of its graph and the xaxis.
Main Activity
Group the students into 10. Give them 30 minutes to work on the given
polynomials. Assign a leader in each group and a reporter to discuss
the output of the group.
Tasks
1. Graph the polynomial function: f(x) = 4x3 + 16x2 + 9x – 9
2. Sketch the graph of the polynomial function: f(x) = x3 + 3x2 - x – 4
Closing Activity
a) Emphasize the key points of the lesson by asking:
1) How do you graph a given polynomial function?
2) How do you find the zeroes of a polynomial function?
b) Summarize their responses as follows:
 To graph a polynomial function, first find the zeroes of the
function. Then construct a table of values of the given variable
and the polynomial function. These values should represent
points in the different intervals into which the zeroes of the
function divide the x-axis. Plot these values on a Cartesian
coordinate plane. If the points are rather far apart, assign
fractional values to the variable to determine the shape of the
curve more accurately.
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Chapter 1: Working with Polynomial Functions
HOMEWORK
1. Graph the function y = x2 + 3x2 –x –3 using (-3,-2,0,1,2,3) as the
domain.
2. Approximate the real zeroes by graphing the polynomial function:
y = x3 – 4 x2 + x + 5
REFERENCES
Foster and Gordon. Algebra 2 With Trigonometry—Applications and
Connections
Jose-Dilao. Advanced Algebra, Trigonometry and Statistics. Functional
Approach. 91–92.
Math IV: Advanced Algebra, Trigonometry and Statistics – 2002. BEC.
Mathematics IV – SEDP Series.
Yu-hico. Experiencing Mathematics.
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Chapter 1: Working with Polynomial Functions
Unit Integration Plan
Objectives
While studying polynomials, students can:

study drug absorption rates;

investigate how volcanic eruptions are predicted;

apply polynomials to situations involving population growth;

explore, analyze, and compare the benefits and liabilities of annuities;

analyze, solve and perform real-world problems.
Procedure
Ask each group to solve the following problems:
1. Given the prospective earnings and the prevailing interest rate,
determine the amount that will be available for graduate school
expenses if a college student saves income from summer
employment following high school, until he/she enters graduate
school after four years of college.
2. Given the sum of the length, width, and depth of carry-on luggage,
determine the real domain for these variables as well as the
maximum volume of a carry-on item if the length is to be 10
centimeters more than the depth.
3. Plot the given data on the number of minutes needed to have
Caucasian skin tanned at different times of the day in Boracay. Use
a graphing calculator to determine a polynomial function to graph
the data. Use the function to determine the time of day when skin
will tan in 30 minutes. Explain why the time would be the same or
different in different parts of the Philippines.
4. Given three patterns for constructing closed rectangular containers
from pieces of cardboard, determine which of the three patterns
produces a container with maximum volume and minimum waste.
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Using the computed values, students can be asked to apply their
mathematical skills in making bags, ornamental boxes and useful
containers which they can sell. They can use any available
resources to produce these products and their creativity in making
the best appropriate design. Accuracy and quality of work are the
main criteria for evaluation.
The products produced may be accompanied by a marketing plan,
so students can sell it later and earn some profit.
5. Students may interview professionals in various fields of study (e.g.,
economics, engineering, architecture, social sciences, etc. to find
out how they make use of polynomials in their work.
As a final project, they may work with these professionals with the
supervision of the teacher or some other adult or student’s
guardian. In this way, students may have more appreciation for
mathematics, and develop work ethics at the same time.
Assessment
In a scale of 6, where 6 is the highest score and 1 is the lowest score,
rate the overall performance of the students in the culminating activity.
6 Exemplary response







gives a complete response with a clear, coherent, unambiguous
and elegant explanation;
includes a clear and simplified diagram;
communicates effectively to the identified audience;
shows understanding of the open-ended problem's
mathematical ideas and processes;
identifies all the important elements of the problem;
may include examples and counterexamples;
presents strong supporting arguments.
5 Competent response

gives a fairly complete response with reasonably clear
explanations;
 may include an appropriate diagram;
 communicates effectively to the identified audience;
 shows understanding of the problem's mathematical ideas and
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processes;
 identifies the most important elements of the problem;
 presents solid supporting arguments.
4 Minor Flaws But Satisfactory

completes the problem satisfactorily, but the explanation may be
muddled;

argumentation may be
inappropriate or unclear;

understands the underlying mathematical ideas;

uses mathematical ideas effectively.
incomplete;
diagram
may
3 Serious Flaws But Nearly Satisfactory

begins the problem appropriately but may fail to complete or
may omit significant parts of the problem;

may fail to show full understanding of mathematical ideas and
processes;

may make major computational errors;

may misuse or fail to use mathematical terms;

response may reflect an inappropriate strategy for solving the
problem.
2 Begins, But Fails to Complete Problem

explanation is not understandable;

diagram may be unclear;

shows no understanding of the problem situation;

may make major computational errors.
1 Unable to Begin Effectively

words do not reflect the problem;

drawings misrepresent the problem situation;

copies parts of the problem but without attempting a solution;

fails to indicate which information is appropriate to then
problem.
Source http://intranet.cps.k12.il.us/Assessments/Ideas_and_Rubrics/Rubric_Bank/MathRubrics.pdf
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be