§ 4.5 Multiplication of Polynomials

Angel, Elementary Algebra, 7ed 2 Multiplying Polynomials To multiply a monomial by a monomial, multiply their coefficients and use the product rule of exponents. a.) (5x 5 )(2x) = 5 · 2 · x 5 ·x = 10x 6 b.)(6x 2 y 2 )(3xy 4 ) = 6 · 3 · x 2 · x · y 2 · y 4 = 18x 3 y 6 a.) 3(x - 4) = 3(x) + 3(-4) = 3x - 12 b.)(-3c 2 + 5c – 6)(-6c) = (-6c)(-3c 2 ) + (-6c)(5c) + (-6c)(-6) = 18c 3 – 30c c To multiply a polynomial by a monomial, use the distributive property.

Angel, Elementary Algebra, 7ed 3 Multiplying Polynomials To multiply two binomials, use the distributive property so every term in one polynomial is multiplied by every term in the other polynomial. Example: a.) (7x + 3)(2x + 4) = (7x + 3)(2x) + (7x + 3)(4) = 14x 2 + 6x + 28x + 12 = 14x x + 12 b.) (z + 2y)(4z – 3) = (z + 2y)(4z) + (z + 2y)(-3) = 4z 2 + 8yz + (-3z) + (-6y) = 4z 2 + 8yz – 3z – 6y A common method used to multiply two binomials is the FOIL method.

Angel, Elementary Algebra, 7ed 4 The FOIL Method Consider (a + b)(c + d): FOILFOIL Stands for the first – multiply the first terms together. (a + b) (c + d): product ac F 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

Angel, Elementary Algebra, 7ed 5 The FOIL Method Using the FOIL method, multiply (7x + 3)(2x + 4). (7x)(2x) = (7x)(2x) (7x + 3)(2x + 4) F F O O + (7x)(4) + (3)(2x) I I + (3)(4) L L 14x x + 6x + 12 = 14x x + 6x + 12 = 14x x + 12

Angel, Elementary Algebra, 7ed 6 Formulas for Special Products Product of the Sum and Difference of Two Terms (a + b)(a – b) = a 2 – b 2 This special product is also called the difference of two squares formula. Example: a.)(7x + 3) (7x – 3) = 49x b.)(z 3 – 2y 4 ) (z 3 + 2y 4 ) = z 6 – 4y 8

Angel, Elementary Algebra, 7ed 7 Formulas for Special Products Square of Binomials (a + b) 2 = (a + b)(a + b) = a 2 + 2ab + b 2 (a – b) 2 = (a – b)(a – b) = a 2 – 2ab + b 2 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.) (5x + 3) 2 = 25x x + 9 b.) (z 3 – 12y) 2 = z 6 – 24yz y 2