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H. Algebra 2
6.1 Polynomial Functions
A monomial is an expression that is either a real number, a variable or a product of real numbers and variables
with whole number exponents.
A polynomial is a monomial or the sum or difference of monomials. Any polynomial is made up of monomial
pieces.
Standard Form of a Polynomial – Descending order that lists monomial terms by decreasing powers
Degree of a Polynomial – largest degree of any term of the polynomial
Write each polynomial in standard form. Then classify it by degree and number of terms.
1. 2 x3  3x  4 x5
2. 2 x3  3x4  2 x3  5
3
2
3
2
4. (6 x  6 x  7 x  1)  (3x  5x  2 x  8)
3. x2  4  8x  2 x3
3
2
2
5. (3x  12 x  5x  1)  ( x  5x  8)
LINEAR
QUADRATIC
CUBIC
QUARTIC
End Behavior: (pg. 306)
Degree is EVEN
Degree is ODD
Leading Coefficient:
POSITIVE
Leading Coefficient:
NEGATIVE
Practice:
Describe the end behavior of each polynomial function.
1. f ( x)  3x  2
2. f ( x)  3x 5  4 x 4
3. f ( x)  2 x 3  5x
4. f ( x)  4 x 2  3x  1
5. f ( x)  x 3  x  5
6. f ( x)  x 6  5x 3  2
Modeling Data with a Polynomial Function
We have already used a linear and quadratic function to model data. Sometimes you can fit data more closely
by using a polynomial model degree of three or greater.
1. Use your calculator to determine whether a linear model, quadratic model, or cubic model best fits the
values in the table. Use the model to predict the value for y when x = 25.
X
Y
0
10.1
5
2.8
10
8.1
15
16
20
17.8
2. The table below shows world gold production for several years. Find a quartic function to model the
data. Use it to estimate production in 1988.
Year
1975 1980
Production
38.5 39.2
(millions of
troy ounces)
1985
49.3
1990
70.2
1995
71.8
2000
82.6
H. Algebra 2
6.2 Polynomial and Linear Factors
Writing a Polynomial in Standard Form
1. Write the expression (x + 1)(x + 2)(x + 3) as a polynomial in standard form.
2. Write the expression (x + 1)(x + 1)(x + 2) as a polynomial in standard form.
3. Write the expression (x – 1)(x + 3)(x + 4) as a polynomial in standard form.
Writing a Polynomial in Factored Form
4. Write 2x3 + 10x2 + 12x in factored form.
5. Write 3x3 – 3x2 – 36x in factored form.
6. Write 3x3 – 18x2 + 24x in factored form.
Applications: Polynomial Functions
Volume of a Box (V = L * W * H)
7. Panda Incorporated has asked its employees to design an open-top box made from a piece of
cardboard that is 36 inches on each side. It is to be made by cutting equal squares in the corners and
turning up the sides.
a. Label the sides using “x” as the box height when the corners are turned up.
b. Write the equation in terms of “x” that represents the volume of the box.
c. Describe a realistic domain.
d. What dimensions of the box will produce a maximum volume?
e. What is the maximum volume?
Relative Maximum: the greatest y-value of the points in a particular region of a graph.
Relative Minimum: the least y-value of the points in a particular region of a graph.
8. Several popular models of carry-on luggage have a length of 10 inches greater than their depth. To
comply with airline regulations, the sum of the length, width and height must not exceed 40 inches.
a. Write a function V for the volume in terms of the depth x.
b. Describe a realistic domain.
c. What is the maximum possible volume of a piece of luggage? What are the corresponding
dimensions?
Factors and Zeros (Roots) of a Polynomial Function
If a polynomial is in factored form, you can use the Zero Product Property to find the values that will make
the equation equal to zero.
9. Find the zeros of f(x) = (x - 2)(x + 1) (x – 3) and graph
10. Find the zeros of f(x) = (x + 3)(x – 4) (x –5) and graph
You can reverse the process and write linear factors when you are given the zeros. This relationship is
described by the factor theorem.
Factor Theorem: The expression x – a is a linear factor of a polynomial if and only if the value a is a zero (root)
of the related polynomial function.
Writing a Polynomial Function from its Zeros
11. Write a polynomial function in standard form with zeros at –2, 3, and 3.
12. Write a polynomial function in standard form with zeros –4, -2 and 1.
13. Write a polynomial function in standard form with zeros –4, -2 and 0.
Multiplicity: equal to the number of times the zero occurs
Repeated zeros (a.k.a repeated root)
Finding the Multiplicity of a Zero
14. Find any multiple zeros of y = x4 + 6x3 + 8x2 and state the multiplicity
15. For each function, find any multiple zeros and state the multiplicity
a. y = (x – 2)2 (x + 1)(x – 3)
b. y = x3 – 4x2 + 4x
H. Algebra 2
6.3 Dividing Polynomials
You can use long division to help find all zeros of a polynomial function.
Reminder: divisor dividend
Find the following quotients BY HAND. Do not use decimals; write remainders.
1. 4 544
2. 12 1640
3. 3 432
Divide: ( x 3  3x 2  4 x  12)  ( x 2  4)
Practice: Find each quotient using long division.
1. (x 3  3x 2  3x  2)  (x 2  x 1)

2. ( x 3  3x 2  13x  15)  ( x 2  2 x  3)
3. ( x 2  4 x  4)  ( x  2)
4. ( x 3  11x 2  39 x  45)  ( x 2  6 x  9)
If your divisor is a linear expression (degree of 1), you can use this method:
Synthetic Division is a method used to divide polynomials.
( x 3  3x 2  4 x  12)  ( x  3)
Find the quotient: (x  x  9x  9)  (x 1)
3
2

Practice: Find each quotient using Synthetic Division.
1.
( x 3  3x 2  4 x  12)  ( x  3)
2.
(x 2  3x  2)  (x 1)

3.
( x 3  x 2  4)  ( x  2)
4. (x  5x 18)  (x  3)
3

2
When you divide by a linear factor (x – r) and there is NO REMAINDER, that linear factor is a factor of the
function; f(r) = 0
When you divide by a linear factor and there is A REMAINDER, then the function is not divisible by that
factor; f(r) ≠ 0
First, determine whether or not the function is divisible by the given factor. If it is, factor the function
completely.
1. f ( x)  x 2  3x  2; x  1
2. f ( x)  x 2  4 x  4; x  2
3. f ( x)  x 3  6 x 2  x  30; x  3
4. f ( x)  x 3  3x 2  6 x  7; x  4
5. f ( x)  x 3  5x 2  18; x  3
6. f ( x)  x 4  5x 2  4 x  12; x  4
Find all zeros of each function. One of the zeros is given.
8. f ( x)  25x 4  40 x 3  19 x 2  2 x ; 
7. f ( x)  5x 3  4 x 2  20 x  16 ; 2
9. f ( x)  10 x 3  41x 2  32 x  20 ;
5
2
10. f ( x)  3x 3  4 x 2  35x  12 ; 3
1
5
6.7 Permutations and Combinations
Example 1 With Replacement- Eight pieces of paper are numbered from 1 to 8 and placed in a box. One
paper is drawn from the box and the number is written down. The piece of paper is replaced in the box. Then
a second piece of paper is drawn. That number is written down. Finally, the numbers are added together.
How many different ways can a total of 12 be obtained?
Example 2 Without Replacement- Eight pieces of paper are numbered from 1 to 8 and placed in a box. Two
pieces of paper are drawn at the same time and the numbers are added together. How many different ways
can a total of 12 be obtained?
The Fundamental Counting Principle
Le E1 and E2 be two events. The first even E1 can occur in m1 different ways. After E1 has occurred, E2 can
occur in m2 different ways. The number of ways that the two events can occur is m 1 X m2
Example- How many different pairs of letters from the English alphabet are possible?
Example- How many different telephone numbers are possible within each area code. (Cannot begin with 0 or
1)
Permutation of n different elements is an ordering of the elements such that one element is first, one is
second, one is third and so on.
Example- How many different permutations are possible for the letters A B C D E F
Number of Permutations of n Elements:
The number of permutations of n elements is
Sometimes you may be interested in ordering a subset of a collection of elements rather than an entire
collection. You may want to choose (and order) r elements out of a collection of n elements. Such an ordering
is called a permutation of n elements taken r at a time.
Example- 8 horses are running in a race. In how many different ways can these horses come in first, second
and third? (assume no ties)
Permutations of n Elements Taken r at a Time
The number of permutations of n elements taken r at a time is
**
Example- In how many distinguishable ways can the letters in BANANA be written?
Combinations
When you count the number of possible permutations of a set of elements, order is important. As a final topic
in this section, you will look at a method of selecting subsets of a larger set in which order is not important.
Such subsets are called combinations of n elements taken r at a time.
Example- {A, B, C} and {B, A, C} are the same because they contain the same three elements. You would count
only one of the two sets.
Example- In how many different ways can three letters be chosen from the letters A B C D and E? (order does
not matter)
Combinations of n Elements Taken r at a Time
The number of combinations of n elements taken r at time is
Example- The standard poker hand consists of five cards dealt from 52 cards. How many different poker
hands are possible?
Example- Find the total number of subsets of a set that has 10 elements
Exercises:
1.Evaluate 6P2 and 6C2
2. A local building company is hiring extra summer help. They need four additional employees to work outside
in the lumber yard and three more to work in the store. In how many ways can these positions be filled if
there are 10 applicants for the outside work and five for the inside work?
3. In how many distinguishable ways can the letters CALCULUS be written?
6.5 The Rational Root Theorem
p
is in simplest form and is a rational root of the polynomial equation
q
an x n  an1 x n1  ...  a1 x  a0  0 with integer coefficients, then p must be a factor of a 0 and q must be a
factor of an.
If
You can use the rational root theorem to find any rational roots of a polynomial equation with integer
coefficients.
Example- Find the rational roots of x3 + x2 - 3x – 3 = 0
Example- Find the rational roots of x3 - 4x2 - 2x + 8 = 0
Example- Using the Rational Root Theorem
Find all roots of 2x3 – x2 + 2x – 1 = 0
Example- Using the Rational Root Theorem
Find all roots of x3 – 2x2 – 5x + 10 = 0
Example- Using the Rational Root Theorem
Find all roots of 3x3 + x2 – x + 1 = 0
6.8 The Binomial Theorem
To expand a binomial being raised to a power, first multiply, then write in standard form:
(a + b)2 =
(a + b)3 =
Pascal’s Triangle: Triangular array of numbers formed by first lining the border with 1’s and then placing the
sum of two adjacent numbers within a row between and underneath the original two numbers:
Using Pascal’s Triangle: Expand (a + b)6
Example- Expand (x – 2)4
Binomial Theorem- For every integer n, (a + b)n =
Example- Using Binomial Theorem- expand (g + h)4
Example- Using Binomial Theorem-expand (v + w)9
Example- Using Binomial Theorem- expand (c – 2)5
Example- Use Binomial Theorem- expand (x – y)8
Example- Use Binomial Theorem- expand (3x – 2y)4