Exponents and Polynomials

Chapter Sections 4.1 – Exponents 4.2 – Negative Exponents 4.3 – Scientific Notation 4.4 – Addition and Subtraction of Polynomials 4.5 – Multiplication of Polynomials 4.6 – Division of Polynomials Chapter 1 Outline

Multiplication of Polynomials

Multiplying Polynomials To multiply two monomials, multiply the coefficients and use the product rule of exponents. Example: (7x3)(6x5) = 7 · x36 · 6 · x5 = 42x8 To multiply a polynomial by a monomial, use the distributive property: a(b + c) = ab + ac Example: Multiply 3x(2x2 + 4) 3x(2x2 + 4) = (3x)(2x2) + (3x)(4) = 6x3 + 12x

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.) (x + 3)(x + 4) = (x + 3)(x) + (x + 3)(4) = x2 + 3x + 4x + 12 = x2 + 7x + 12 A common method used to multiply two binomials is the FOIL method.

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 (2x - 3)(x + 4) . + (-3)(4) L = (2x)(x) (2x - 3) (x + 4) F + (-3)(x) I O + (2x)(4) = 2x 2 + 8x - 3x - 12 = 2x 2 + 5x - 12

Formulas for Special Products Product of the Sum and Difference of the Same Two Terms (a + b)(a – b) = a2 – b2 The expression on the right side of the equals sign is called the difference of two squares. Example: a.) (x + 5) (x – ) = x2 - 25

Formulas for Special Products 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.) (x + 5)2 = (x)2 + 2(x)(5) + (5)2 = x2 + 10x + 25