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

Multiplication of Polynomials § 5.2 Multiplication of Polynomials

Multiply a Monomial by a Polynomial To multiply polynomials, you must remember that each term of one polynomial must be multiplied by each term of the other polynomial. To multiply monomials, we use the product rule for exponents. Product Rule for Exponents

Multiply a Monomial by a Polynomial Example:

Multiply a Monomial by a Polynomial When multiplying a monomial by a polynomial that contains more than two terms we can use the expanded form of the distributive property. Distributive Property, Expanded Form

Multiply a Monomial by a Polynomial Example:

The FOIL Method F O I L Consider (a + b)(c + d): Stands for the first – multiply the first terms together. (a + b) (c + d): product ac Stands for the outer – multiply the outer terms together. (a + b) (c + d): product ad O Stands for the inner – multiply the inner terms together. (a + b) (c + d): product bc I Stands for the last – multiply the last terms together. L (a + b) (c + d): product bd The product of the two binomials is the sum of these four products: (a + b)(c + d) = ac + ad + bc + bd.

The FOIL Method Using the FOIL method, multiply (7x + 3)(2x + 4). + (3)(4) L = (7x)(2x) (7x + 3)(2x + 4) F + (3)(2x) I O + (7x)(4) = 14x 2 + 28x + 6x + 12 = 14x 2 + 34x + 12

Find the Square of a Binomial Square of Binomials (a + b)2 = (a + b)(a + b) = a2 + 2ab + b2 (a – b)2 = (a – b)(a – b) = a2 – 2ab + b2 To square a binomial, add the square of the first term, twice the product of the terms and the square of the second term. Example: a.) (3x + 7)2 = 9x2 + 42x + 49 b.) (4x2 – 5y)2 = 16x4 – 40x2y+ 25y2

Product of the Sum and Difference The Product of the Sum and Difference of Two Terms (a + b)(a – b) = a2 – b2 This special product is also called the difference of two squares formula. Example: a.) (2x + 3y) (2x – 3y) = 4x2 – 9y2 b.) (3x + 4/5) (3x – 4/5) = 9x2 – 16/25