1. Polynomials and Polynomial Functions
*Chapter 5
Polynomials and Polynomial Functions

2. Chapter Sections 5.1 – Addition and Subtraction of Polynomials
5.2 – Multiplication of Polynomials
5.3 – Division of Polynomials and Synthetic Division
5.4 – Factoring a Monomial from a Polynomial and Factoring by Grouping
5.5 – Factoring Trinomials
5.6 – Special Factoring Formulas
5.7-A General Review of Factoring
5.8- Polynomial Equations
Chapter 1 Outline

3. 5.1 Addition and Subtraction of Polynomials
§ 5.1
5.1 Addition and Subtraction of Polynomials
1. Identify the coefficient and degree of a monomial.
3. Classify polynomials.
4. Identify the degree of a polynomial.
5. Evaluate polynomials.
6. Write polynomials in descending order of degree.
Combine like terms:
1. Add polynomials.
2. Subtract polynomials.

4. Recall: If a and b are real numbers and then
Exponents
Recall:
If a and b are real numbers and then

5. Example 1: Evaluate each exponential form.
Simplify.
a. b. c.
d. e. f.
Solution a. b. c.

6. Example 2:
Simplify.
a. b c d E.
Solution a. b.

7. Find the Degree of a Polynomial
A polynomial is a finite sum of terms in which all variables have whole number exponents and no variable appears in a denominator.
3x2 + 2x + 6 is a polynomial in one variable x
x2y – 7x + 3 is a polynomial in two variables x and y
x1/2 is not a polynomial because the variable does not have a whole number exponent
The degree of a term of a polynomial in one variable is the exponent on the variable in that term.
Example: 5x x x
Degree: , , 0
The degree of a polynomial is the same as that of its highest-degree term.
Example: 5x6 + 4x3 – 7x + 9
(Sixth)

8. Leading Term & Coefficient of a Polynomial
The leading term of a polynomial is the term of highest degree. The leading coefficient is the coefficient of the leading term.
Example: 2x5 – 3x2 + 6x – 9
Degree: 5
The leading term: 2x5
The leading coefficient: 2
Example:
5x6 + 4x3 – 7x + 9
A polynomial is written in descending order (or descending powers) of the variable when the exponents on the variable decrease from left to right.
A polynomial with one term is called a monomial.
A binomial is a two-termed polynomial.
A trinomial is a three-termed polynomial.

9. a. 4ab2 is a monomial because it has a 1 term.
Example 3:
Determine whether the expression is a monomial, binomial, trinomial, or none of these. Identify the degree.
a. 4ab2 b. –9x5 + z c. 4n7+ 2n – 1 d. x3 + 9x2 – x e. 8𝑦− 6 𝑥
Answer
a. 4ab2 is a monomial because it has a 1 term.
b. –9x5 + z is a binomial (2 terms), Degree 5
c. 4n7+ 2n – 1 is a trinomial (3 terms), Degree 7
d. x3 + 9x2 – x + 4 is a polynomial with no special name, Degree 3
8𝑦− 6 𝑥 is not a polynomial, why?

10. Adding & Subtracting Polynomials: Combine like terms Example 4:
Add and write the resulting polynomial in descending order of degree.
Solution
To add polynomials, remove parentheses if any are present. Then combine like terms (Terms having the same variable/s & same exponent).

11. Example 5: Add or Subtract.
Solution

12. Example 6: Add or Subtract. (Combine like terms.)
Solution

13. Example 7:
Write an expression in simplest form for the perimeter of the rectangle shown.
9b – 10
5b + 2

14. Evaluate Polynomial Functions
A polynomial function is an expression used to describe the function in a polynomial.
Example 8:
Given the polynomial function P(x) = 4x3 – 6x2 – 2x + 9,
find P(0) & P(2).
P(0) = 4(0)3 – 6(0)2 – 2(0) + 9 = 9
P(2) = 4(2)3 – 6(2)2 – 2(2) + 9
= 4(2)3 – 6(2)2 -2(2) + 9 =
= 32 – 24 – = 13

15. Understand Graphs of Polynomial Functions