Download Definition Sheet

Definition Sheet
Polynomial- An expression which is the sum of the form
axk + axk-1 +…+ ax + a where k is a positive integer and a is any coefficient.
Standard Form-the terms of a polynomial are placed in descending order, from
largest degree to smallest degree.
Degree- is the exponent of the variable of each term in a polynomial
Degree of a Polynomial- the largest degree of its terms
Leading Coefficient- the coefficient of the first term when a polynomial is written
in standard form
Binomial- a polynomial with two terms
Trinomial- a polynomial with three terms
FOIL- pattern used to multiply two binomials: Multiply the First, Outer, Inner,
and Last terms.
Group Semester Unit Project
NCTM Standards: Grades 9-12:
 Algebra:
o Represent and analyze mathematical situations and structures using algebraic
symbols:
 write equivalent forms of equations, inequalities, and systems of
equations and solve them with fluency—mentally or with paper and
pencil in simple cases and using technology in all cases
o Understand patterns, relations, and functions
 understand and perform transformations such as arithmetically
combining, composing, and inverting commonly used functions, using
technology to perform such operations on more-complicated symbolic
expressions
 understand and compare the properties of classes of functions,
including exponential, polynomial, rational, logarithmic, and periodic
functions
 interpret representations of functions of two variables
 Problem Solving:
o build new mathematical knowledge through problem solving
o solve problems that arise in mathematics and in other contexts
o apply and adapt a variety of appropriate strategies to solve problems
o monitor and reflect on the process of mathematical problem solving
 Communication
o organize and consolidate their mathematical thinking through communication;
o communicate their mathematical thinking coherently and clearly to peers,
teachers, and others
 Connections
o recognize and use connections among mathematical ideas;
o understand how mathematical ideas interconnect and build on one another to
produce a coherent whole;
o recognize and apply mathematics in contexts outside of mathematics
 Representations
o create and use representations to organize, record, and communicate
mathematical ideas
SOLs:
A. 11 The student will add, subtract, and multiply polynomials and divide polynomials with
monomial divisors, using concrete objects, pictorial and area representations, and
algebraic manipulations.
AII.3 The student will
a) add, subtract, multiply, divide, and simplify radical expressions containing positive
rational numbers and variables and expressions containing rational exponents
10.1 Adding and Subtracting Polynomials
Objective: For students to understand how to add and subtract polynomials along with standard
definitions.
Materials: paper, pencil, McDougal Littell 2004 Algebra 1 textbook, algeblocks, homework
handout, elmo
Interdisciplinary: History and social issues
References: http://standards.nctm.org/
http://www.pen.k12.va.us/VDOE/Superintendent/Sols/home.shtml
McDougal Little Algebra 1 Book, 2004 Edition
Lesson Plan:
 Use elmo to give students a visual on how to use algeblocks
o Use the Basic Mat –identify x^3, x^2, x, and units blocks. Show how to add
and subtract on the mat.
 Hand out the Definitions handout explaining that these are terms that will be used in
the next few lessons
 Divide class into groups of two or three giving each group a set of Algeblocks
allowing students to get a hands-on experience while communicating with others.
 Take one x block and one units block and place it side by side on the positive part of
the mat. Right below that place 3 x blocks and 4 units blocks side by side in the
positive section. Now add these together.
o Teacher would walk around seeing if anyone needs help or has any questions.
o The result would be that they have 4 x blocks and 5 units blocks and make
sure everyone got that answer and if not try and figure out, with the student,
what happened by asking questions.
o Demonstrate the process on the elmo
o Ask “How do you think you would say this answer?” The students would
think about it and probably throw out some random answers but should easily
be guided to say something in the area of 4x + 5 or 5 and 4x.
o Say, “This is how it would be written 4x + 5,” and write that on the board.
 Next tell them to place an x^3 block in the positive section along with 2 x^2 blocks,
and 1 units block and 3 x blocks in the negative section all in one row. Then tell them
to place right below the others, in a row, an x^2 block, and 2 units blocks in the
negative part and an x^3 block and an x in the positive part. Then tell them to add
them together.
o Repeat the steps above except the result would be 2 x^3 blocks, 1 x^2 block, 2
x blocks that are negative and a 1 units block that is negative.
o They should hopefully respond this time saying something like 2x^3 + x^2 –
2x – 1.
 Say, “Now we are going to demonstrate what we just did with the Algeblocks using
paper and pencil on the board”
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What we just did was add together x^3 + 2x^2 – 3x +1 and x^3 – x^2 + x – 2.(write
two polynomials beside each other on the board)
Say, “These are called polynomials. A polynomial is the addition of expressions of
the form ax^n where a is the coefficient and n is the exponent.”
When writing a polynomial you want to put it into standard form where the
expressions are written in descending order from the largest degree to the smallest
degree, which is what we have done.
A degree is the exponent of the variable in each expression and the degree of a
polynomial is the highest degree of all the terms. Ask what would the degree of this
polynomial be. The class would answer 3. The leading coefficient of the polynomial
is the coefficient of the first term when the polynomial is in standard form.
In order to add or subtract polynomials, you have to line them up in columns with
their matching degree. Teacher will demonstrate on board. Then add or subtract each
coefficient of the expression in its column.
On board will do the example 2x^2 + x – 5 subtracting x + x^2 + 6. Ask class to tell
what the steps are: first put in standard form then line up the degrees then subtract
each expression resulting x^2 + 0x – 11 = x^2 – 11.
We will then give them a word problem saying, “From 1890 through 1990, the
number of men M and women W in the United States labor force can be modeled by
the following equations, where t is the number of years since 1890. Number of men
in millions: M = 0.0016t^2 + 0.315t + 19.467 Number of women in millions: W =
0.007t^2 – 0.228t + 5.908 Find a model for the total number S of men and women in
the U.S. labor force.
We will demonstrate on the board with students telling the teacher each step.
Answer: S = 0.0086t^2 + 0.087t + 25.375
Then we will assign each group a problem from the guided practice problems
beginning with 13-18 in their text book on page 579 and have each group present it to
the class.
Walk around and help students if needed and ask questions to help them along.
Distribute homework
Timeline:
 Explanation of Algeblocks – 5 minutes
 2 Algeblocks problems – 15-20 minutes
 Explanation of last problem along with definitions – 10-15 minutes
 2 Example problems on board – 10-15 minutes
 Group work and Demonstrations – 20-25 minutes
Student Response: Students would probably like working with the Algeblocks and they
would probably help them to understand how to add expressions. Students may have
trouble realizing how to match up polynomials according to their degree. They may also
make the common mistake of forgetting to change signs correctly when subtracting from
one another and also classifying polynomials by their degree or number of terms. All in
all I think it is a clear cut lesson that will be easily understood.
Homework
Identify the leading coefficient and find the degree of the polynomial.
1. -4x2 + 2x – 1
2. 8 + 5y2 – 3y
3. -6
Write the polynomial in standard form.
4. 7x + 9 – 4x3 + x2
5. 3 – 5x6 + 2x + 4x2
Add or Subtract the Polynomials
6. (x2 – 7) + (2x2+2)
7. (-2x4 + 6x2 + 5) – (-2x4 + 5x2 + 1)
8. (-3x2 + 5) + (-x2 + 4x – 6)
9. (x3 + x2 + 1) – x2
10. (x3-6x) – (2x3 + 9) – (4x2 + x3)
11. From 1989-1993 the amounts (in billions of dollars) spent on natural gas N and
electricity E by the U.S. residents can be modeled by the following equations where t is
the number of years since 1989.
Gas spending model: N = 1.488t2 – 3.403t + 65.590
Electricity spending model: E = -0.107t2 + 6.897t + 169.735
Find a model for the total amount, A, (in billions of dollars) spent on natural gas and
electricity by the U.S. residents from 1989-1993.
Homework Solutions
Identify the leading coefficient and find the degree of the polynomial.
1. -4x2 + 2x – 1
2. 8 + 5y2 – 3y
3. -6
-4; 2
5; 2
-6; 0-6x0
Write the polynomial in standard form.
4. 7x + 9 – 4x3 + x2
5. 3 – 5x6 + 2x + 4x2
3
2
-4x + x + 7x + 9
-5x6 + 4x2 + 2x + 3
Add or Subtract the Polynomials
6. (x2 – 7) + (2x2+2)
7. (-2x4 + 6x2 + 5) – (-2x4 + 5x2 + 1)
x2 – 7
-2x4 + 6x2 + 5
2
+ 2x + 2
− -2x4 + 5x2 + 1
3x2 – 5
x2 + 4
8. (-3x2 + 5) + (-x2 + 4x – 6)
-3x2 + 0x + 5
+ -x2 + 4x – 6
-4x2 + 4x – 1
9. (x3 + x2 + 1) – x2
x3 + x2 + 1
− 0x3 + x2 + 0
x3 + 2x2 + 1
10. (x3-6x) – (2x3 + 9) – (4x2 + x3)
First distribute the negatives into the last two polynomials then put each polynomial in
standard form.
x3 + 0x2 – 6x + 0
-2x3 + 0x2 + 0x – 9
+ -x3 – 4x2 + 0x + 0
-2x3 – 4x2 – 6x – 9
11. From 1989-1993 the amounts (in billions of dollars) spent on natural gas N and
electricity E by the U.S. residents can be modeled by the following equations where t is
the number of years since 1989.
Gas spending model: N = 1.488t2 – 3.403t + 65.590
Electricity spending model: E = -0.107t2 + 6.897t + 169.735
Find a model for the total amount, A, (in billions of dollars) spent on natural gas and
electricity by the U.S. residents from 1989-1993.
1.488t2 – 3.403t + 65.590
+ -0.107t2 + 6.897t + 169.735
A = 1.381t2 + 3.494t + 235.325
10.2 Multiplying Polynomials
Objective: For students to understand how to multiply polynomials along with standard
definitions.
Materials: paper, pencil, McDougal Littell 2004 Algebra 1 textbook, algeblocks, homework
handout, elmo
Interdisciplinary: social issues
References: http://standards.nctm.org/
http://www.pen.k12.va.us/VDOE/Superintendent/Sols/home.shtml
McDougal Little Algebra 1 Book, 2004 Edition
Lesson Plan:
 Use elmo to give students a visual on how to use algeblocks
o Use the Quadrant Mat –identify x^3, x^2, x, and units blocks. Show how to
multiply on the mat.
 Tell the students to refer to their definition sheet.
 Divide class into groups of two or three giving each group a set of Algeblocks
allowing students to get a hands-on experience while communicating with others.
 We are going to use the polynomials we used yesterday in today’s lesson.
 Place the polynomial x + 1 flat within the horizontal positive section of the mat.
Place the polynomial 3x + 4 flat within the vertical positive section. Now try and
multiply these.
o Teacher would walk around seeing if anyone needs help or has any questions.
o The result would be that they have 3x^2 + 7x +4 and make sure everyone got
that answer and if not try and figure out, with the student, what happened by
asking questions.
o Demonstrate the process on the elmo
 Next tell them that we are going to solve (x – 2)(-2x + 1). Ask the class how they
think they should set up the problem with the blocks. We should conclude to place
the x on the horizontal positive section and 2 units on the horizontal negative section.
And place the 2 x’s on the vertical negative section and 1 unit on the vertical positive
section. Or vise versa. Tell the class to multiply them.
o Teacher would walk around seeing if anyone needs help or has any questions
again.
o The result would be that they have -2x^2 + 5x -2 and make sure everyone got
that answer and if not try and figure out, with the student, what happened by
asking questions.
o Demonstrate on the elmo.
 Now we are going to learn to do this with pencil and paper by learning some methods
and terms.
 Write the polynomial (x + 4)(x + 5) and the polynomial (x^2 + 20). Then take a poll
and see how many students think the two are equal. Ask how you can decide whether
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they are equal or not. We can use substitution by putting in a value for x where x is
not equal to 0. We will make a table of different solutions for the different x values.
Ex.: use x = 1. we would get (1 + 4)(1 + 5) = 30. and 1^2 + 20 = 21 and 21 is not
equal to 30.
I will show the most important method for multiplying binomials. A binomial is a
polynomial with two terms. The method I will demonstrate is the FOIL method –
Product of the First terms plus the product of the Outer terms plus the product of the
Inner terms plus the product of the Last terms. A good way to remember this is with
cookies and milk.
So lets look back at our “tabled” problem. What is the first step of multiplying (x +
4)(x + 5)? Which is the “First terms” x * x = x^2. What is the next step? The Outer
terms: x * 5 = 5x. What is the next step? The Inner terms: 4 * x = 4x. What is the
Last step? The Last terms: 4 * 5 = 20. What terms do we have in common? 4x and
5x so added together they become 9x. So what is the solution? x^2 + 9x + 20.
Now do this example on your own with their group: (x + 2)(2x – 3)
o Walk around and see if anyone needs help if so try and ask questions to walk
them through it.
o The final solution would be 2x^2 + x – 6.
Now we will try and expand the FOIL/cookie method with a binomial and a
trinomial. Ask if a binomial has two terms how many terms do you think a trinomial
will have? 3.
We are going to extend the cookie method by adding a napkin.
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Lets try (x – 2)(-x^2 + 3x + 5). Ask the students questions to help them guide you
through the problem with solution –x^3 + 5x^2 – x – 10.
 Tell them to try (4x^2 – 3x – 1)(2x – 5) on their own with their group.
o Walk around and see if anyone needs help if so try and ask questions to walk
them through it.
o The final solution would be 8x^3 – 26x^2 + 13x + 5
 Ask if there are any questions about what we went over today. This math is used
continuously as you go through school and any type of carpentry or landscaping will
have to use this type of math.
 Distribute homework.
Timeline:
 Explanation of Algeblocks – 5 minutes
 2 Algeblocks problems – 15-20 minutes
 Table – 5 minutes
 Foil and problems and definitions – 20-25 minutes
Student Response: We feel this is a good method for learning to multiply polynomials.
We think it helps the understanding to do a visual/hands-on method, like with the
Algeblocks, first then to use the written method. We also feel that the cookies analogy
for FOIL will help them to better remember and relate the process. The main confusion
might occur when they have to remember when to add or multiply terms in the FOIL
method.
Homework
1. (2x – 5)(-4x)
2. (2d + 3)(3d + 1)
3. (4k – 1)(3k + 8)
4. (x – 9)(2x + 15)
5. (2.5x – 6.1)(x + 4.3)
6. (a2 + 8)(a2 – a – 3)
7. (x + 6)(x2 – 6x – 2)
8. (3s2 – s – 1)(s + 2)
9. (4w2)(-2w2 – w + 3w3)
10. A football field’s dimensions can be represented by a width of (1/2x + 10) ft. and a
length of (5/4x – 15)ft.
a. Find an expression for the area A of a football field and give your answer as a
trinomial.
Homework Solutions
1. (2x – 5)(-4x)
Distributive property
-8x2 + 20x
4. (x – 9)(2x + 15)
FOIL
2x2 – 3x – 135
7. (x + 6)(x2 – 6x – 2)
2. (2d + 3)(3d + 1)
3. (4k – 1)(3k + 8)
FOIL
FOIL
6d2 + 11d + 3
5. (2.5x – 6.1)(x + 4.3)
FOIL
2.5x2 + 4.65x – 26.23
8. (3s2 – s – 1)(s + 2)
EX. FOIL
EX. FOIL
x3 – 38x – 12
3s3 + 5s2 – 3s -2
12k2 + 29k – 8
6. (a2 + 8)(a2 – a – 3)
EXTENDED FOIL
a4 – a3 + 5a2 – 8a - 24
9. (4w2)(-2w2 – w + 3w3)
EX. FOIL
12w5 – 8w4 – 4w3
10. A football field’s dimensions can be represented by a width of (1/2x + 10) ft. and a
length of (5/4x – 15)ft.
a. Find an expression for the area A of a football field and give your answer as a
trinomial.
A = length * width = (1/2x + 10)ft.* (5/4x – 15)ft.= 5/8x2 + 5x – 150 by FOIL.