Chapter 9 Polynomial Functions

The last functions chapter

Section 9-1 Polynomial Models•A polynomial in x is an expression of the form

where n is a nonnegative integer and • The degree of the polynomial is n• The leading coefficient is• When the exponents are in descending order,

the polynomial is in standard form• Ex1. • A) write in standard form• B) what is the degree?• C) what is the leading coefficient?

1 21 2 1 0...n n n

n n na x a x a x a x a

0na

na

3 2200 50 25 20x x x

•A polynomial function is one whose rule can be written as a polynomial

•A monomial has one term (i.e. 5xyz)•A binomial has two terms (i.e. 5x + 3z)•A trinomial has three terms (i.e. 5x – 3y +

4z)•Open your book to page 558 and look at Ex2.•To find the degree of a polynomial, find the

degree of each individual monomial and use the largest number

•Ex2. Find the degree of

2 3 2 65 6 2 3x y xy y x y

Section 9-2 Graphs of Polynomial Functions

•When discussing the maximum and minimum values, they are speaking in terms of y-values

•The maximum and minimum values are known as the extreme values or extrema

•Relative extrema are the maximum and/or minimum values within a restricted domain▫They are also called the “turning points” of the

function• Open your book to page 564 and look at the

graphs

•The x-intercepts of a function are also known as the roots or the zeros of the function

•To find the exact roots of a quadratic function (degree of 2), use the quadratic function and simplified radical form▫Otherwise you can use your graphing

calculator or factoring to solve• No general formulas for finding the zeros of

polynomial functions of degree higher than 4 exist• For functions of degree 3 or 4, graph and use

your calculator to find the zeros

•A function is considered to be positive on a given interval when the values of the dependent variable (y-values) are positive

•A function is considered to be negative on a given interval when the values of the dependent variable (y-values) are negative

•A function is said to be increasing on an interval if the slope is positive in that interval

•A function is said to be decreasing on an interval if the slope is negative in that interval

•Open your book to page 566 and read the example

Section 9-3 Finding Polynomial Models•Ex1. f(x) = 3x + 2

▫A) find f(0), f(1), f(2), f(3), and f(4)▫B) find the differences between each term

(right – left)• Ex2. f(x) = x² + 4x – 6• A) find f(0), f(1), f(2), f(3), f(4), and f(5)• B) find the first and second differences

• A constant difference will occur at the degree of the polynomial (first differences were the same for a linear function, second differences for quadratic function, third differences for a cubic, etc.)

•Polynomial Difference Theorem: The function y = f(x) is a polynomial function of degree n if and only if, for any set of x-values that form an arithmetic sequence, the nth difference of corresponding y-values are equal and non-zero

•You can use this, along with your calculator to create polynomial models

•Read about coefficients and tetrahedral numbers

•Ex3. a) find the degree of the polynomial b) find the equation for f(x)

x 1 2 3 4 5 6 7 8 9

f(x) -3 24 91 216 417 712 1119 1656 2341

Section 9-4 Division and the Remainder Theorem•Terminology reminder: 100 ÷ 20 = 5

▫100 is the dividend, 20 is the divisor, 5 is the quotient

• Use the algorithm for long division of numbers with polynomials• Ex1. (2x² – 9x – 18) ÷ (x – 6)• Ex2. (6x³ – 7x² + 9x + 4) ÷ (3x + 1)• You must show work on these types of

questions• If there is a remainder, write it:

remainderquotient

divisor

•The remainder must have a smaller degree than the divisor

•Ex3. (4x³ – 4x + 8) ÷ (2x + 6)•If a polynomial, f(x) is divided by x – c, then

the remainder is f(c)•Ex4. Find the remainder of

•You don’t have to do the division to find the remainder. For example 4, just find f(5)

4 26 3 5

5

x x

x

Synthetic Division (Pre-Calc 4-5)•Another way to divide polynomials is with

synthetic division•To use synthetic division, you must write all

polynomials in standard form and include all terms

•To understand synthetic division, you must first be able to use synthetic substitution

•Ex1. Find f(4) for•Synthetic substitution will find the answer to

example 1 in another way

4 3( ) 2 7 8f x x x x

•Synthetic substitution:▫Write the x value to the left▫Write the coefficients to each of the terms in

order to the right (slightly spaced apart)▫Bring down the first coefficient, multiply it by

the x value▫Write the answer under the next coefficient

and add the two together▫Repeat until you have used all of the

coefficients▫The final value is the f(x) value

• Ex2. Solve Ex1. using synthetic substitution

•Ex3. Use repeated factoring to write the polynomial from Ex1. in nested form

•The other numbers below the addition line in Ex2. turn out to be the coefficients to the terms you get when you divide the polynomial by (x – the value)

•So using example 1:

•Look at the top of page 251 from the Pre-Calc book to see how this looks with variables

•This is synthetic division

4 3 3 2 602 7 8 4 2 4 17

4x x x x x x x

x

•Use synthetic division to divide•Ex4.

•Ex5.

3 23 4 2 5 ( 7)x x x x

4 32 9 2 712

x x x

x

Section 9-5 The Factor Theorem•For a polynomial f(x), a number c is a solution

to f(x) = 0 if and only if (x – c) is a factor of f•Factor-Solution-Intercept Equivalence

Theorem: For any polynomial f, the following are logically equivalent statements: 1) (x – c) is a factor of f 2) f(c) = 0

3) c is an x-intercept of the graph y = f(x) 4) c is a zero of f 5) the remainder when f(x) is divided by (x – c) is 0

•Ex1. Factor•Because a term from example 1 can be

factored by 2 (k = 2), that 2 could have been applied to any of the binomials

•Ex2. Find two equations for a polynomial function with zeros:

•You cannot determine the degree because k is unknown

•Open your book to pages 586-587 to see example 2

3 22 8 22 60x x x

2 53, ,3 6and

Section 9-6 Complex Numbers•Imaginary numbers:•Therefore i² = -1•Ex1. Solve without a calculator•Complex numbers: a + bi where a is the real

part and b is the imaginary•The complex conjugate of a + bi is a – bi •Ex2. x = 2 + 3i and y = 5 – 2i

▫A) find x + y▫B) find x – y▫C) find xy

1 i

36

•If a and b are real numbers, then•Without imaginary numbers, you could only

factor the difference of squares (not sum of squares)

•Ex3. Factor x² + 100•Ex4. Write in a + bi form:

•Ex5. Solve: x² – 6x + 20 = 0•Discriminant: b² – 4ac•If the discriminant is:

▫positive, then there are 2 real roots▫negative, then there are 2 complex conjugate

roots▫zero, then there is 1 real root

2 2 ( )( )a b a bi a bi

2 3

5 2

i

i

Section 9-7 The Fundamental Theorem of Algebra

•Fundamental Theorem of Algebra: If p(x) is any polynomial of degree n > 1 with complex coefficients, then p(x) = 0 has at least one complex zero

•A polynomial of degree n has at most n zeros•The multiplicity (number of times the same

zero occurs for a function) of a zero r is the highest power of (x – r) that appears as a factor of that polynomial

•For multiplicity, see Ex1. on page 598

•A polynomial of degree n > 1 with complex coefficients has exactly n complex zeros, if multiplicities are counted

•Let p(x) be a polynomial of degree n > 1 with real coefficients▫The graph of p(x) can cross any horizontal line

y = d at most n times• When given a graph, to find the lowest

degree possible of the equation, draw a horizontal line where it would hit the graph the highest number of times• That is the lowest possible degree (see page

599)

•To verify a number is a zero, plug it in for x and the result should be 0

•If an imaginary number is a zero of a function, so is its complex conjugate

•See example 3 part b to see how to find the remaining zeros

Section 9-8 Factoring Sums and Differences of Powers

•Ex1. Find all of the cube roots of 64•Sums and Differences of Cubes Theorems:

For all x and y,

•These are now factored completely•Ex2. Factor 27c³ – 125d³•Open your book to page 605 to see how to

use the previous theorem for all odd powered functions

•Ex3. find all the zeros of 3x³ – 75x

3 3 2 2

3 3 2 2

( )( )

( )( )

x y x y x xy y

x y x y x xy y

Section 9-9 Advanced Factoring Techniques

•Use chunking (grouping) as another way of factoring

•Ex1. Factor x³ + 2x² – 9x – 18 •Find common factors to factor out and then

use algebra on the remaining terms to simplify

•Not all questions will be factored or chunked in the same way

•Sometimes (with trinomials) it is helpful to separate the middle term in to two separate terms for factoring

•Ex2. Factor 4x² + 4x – 15 •When a polynomial in 2 variables is equal to

0, grouping terms may help you graph it (two separate graphs are created and joined as one)

•Ex3. Draw a graph of y² + xy + 3x + 3y = 0

Section 9-10 Roots and Coefficients of Polynomials

•For the quadratic equation x² + bx + c = 0, the sum of the roots is -b and the product of the roots is c

•Ex1. Find two numbers whose sum is 20 and whose product is 50. Show work.

•For the cubic equation x³ + bx² + cx + d = 0, the sum of the roots is –b, the sum of the products of the roots two at a time is c, and the product of the three roots is -d

•If a polynomial equation p(x) = 0 has leading coefficient and n roots then p(x) =

•For the polynomial equation the sum of the roots is -a, the sum of the products of the roots two at a time is a2, the sum of the products of the roots three at a time is -a3, …, and the product of all the roots is

1na 1 2, ,..., nr r r

1 2 ... nx r x r x r

1 21 2 1... 0n n n

n nx a x a x a x a

n

n

a if n is even

a if n is odd