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

Addition and Subtraction of Polynomials § 5.1 Addition and Subtraction of Polynomials

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

Identifying Polynomials The degree of a term of a polynomial in one variable is the exponent on the variable in that term. Example: 5x6 (Sixth) 4x3 (Third) 7x (First) 9 (Zero) The degree of a polynomial is the same as that of its highest-degree term. Example: 5x6 + 4x3 – 7x + 9 (Sixth)

Find the Degree 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 The degree of the polynomial is 5, the leading term is 2x5 and the leading coefficient is 2.

Identifying Polynomials A polynomial is written in descending order (or descending powers) of the variable when the exponents on the variable decrease from left to right. Example: 5x6 + 4x3 – 7x + 9 A polynomial with one term is called a monomial. A binomial is a two-termed polynomial. A trinomial is a three-termed polynomial.

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

Understand Graphs of Polynomial Functions These graphs have a positive leading coefficient, and therefore, the function continues to increase to the right of some value of x.

Understand Graphs of Polynomial Functions These graphs have a negative leading coefficient, and therefore, the function continues to decrease to the right of some value of x.

Adding Polynomials To add polynomials, remove parentheses if any are present. Then combine like terms. Example:

Subtracting Polynomials Use the distributive property to remove parentheses. (This will have the effect of changing the sign of every term within the parentheses of the polynomial being subtracted.) –(4x3 + 5x2 – 8) = – 4x3 – 5x2 + 8 Combine like terms. Example: (5x – 6) – (2x – 3) = 5x – 6 – 2x + 3 = 3x – 3