1. Chapter 9 Polynomial Functions
The last functions chapter

2. 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?

3. 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

4. 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

5. 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

6. 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

7. 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.)

8. Read about coefficients and tetrahedral numbers
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

9. 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:

10. 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)

11. 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

12. 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

13. Ex3. Use repeated factoring to write the polynomial from Ex1
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

14. Use synthetic division to divide
Ex4.
Ex5.

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

16. 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 to see example 2

17. 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

18. 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

19. 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

20. Let p(x) be a polynomial of degree n > 1 with real coefficients
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)

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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