Polynomials and Polynomial Functions *Chapter 5 Polynomials and Polynomial Functions

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

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.

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

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

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

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: 5x6 4x3 7x 9 Degree: 6 3, 1, 0 The degree of a polynomial is the same as that of its highest-degree term. Example: 5x6 + 4x3 – 7x + 9 (Sixth)

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.

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 + 4 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?

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).

Example 5: Add or Subtract. Solution

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

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

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 – 4 + 9 = 13

Understand Graphs of Polynomial Functions