CHAPTER 4 Polynomials: Operations Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 4.1Integers as Exponents 4.2Exponents and Scientific Notation 4.3Introduction to Polynomials 4.4Addition and Subtraction of Polynomials 4.5Multiplication of Polynomials 4.6Special Products 4.7Operations with Polynomials in Several Variables 4.8Division of Polynomials

OBJECTIVES 4.5 Multiplication of Polynomials Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. aMultiply monomials. bMultiply a monomial and any polynomial. cMultiply two binomials. dMultiply any two polynomials.

4.5 Multiplication of Polynomials Multiplying Monomials Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. To find an equivalent expression for the product of two monomials, multiply the coefficients and then multiply the variables using the product rule for exponents.

EXAMPLE a) (6x)(7x) = 4.5 Multiplication of Polynomials a Multiply monomials. AMultiply: (continued) Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. (6  7) (x  x) = 42x 2

EXAMPLE b) (5a)(  a) = 4.5 Multiplication of Polynomials a Multiply monomials. AMultiply: (continued) Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. (5a)(  1a) = (5)(  1)(a  a) =  5a 2

EXAMPLE c) (  8x 6 )(3x 4 ) = =  24x Multiplication of Polynomials a Multiply monomials. AMultiply: Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. (  8  3) (x 6  x 4 ) =  24x 6 + 4

EXAMPLE a) x(x + 7) = = x 2 + 7x 4.5 Multiplication of Polynomials b Multiply a monomial and any polynomial. BMultiply: Slide 8 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. (x  x)+ (x  7)

EXAMPLE b) 6x(x 2  4x + 5) = = 6x 3  24x x + (6x)( – 5) + (6x)( – 4x) (6x)(x 2 ) 4.5 Multiplication of Polynomials b Multiply a monomial and any polynomial. BMultiply: Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

4.5 Multiplication of Polynomials Multiplying a Monomial and a Polynomial Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. To multiply a monomial and a polynomial, multiply each term of the polynomial by the monomial.

EXAMPLE Solution 5x 2 (x 3  4x 2 + 3x  5) = (5x 2 )(x 3 ) + (5x 2 )(– 4x 2 ) + (5x 2 )(3x) + (5x 2 )( – 5) = 5x 5  20x x 3  25x Multiplication of Polynomials b Multiply a monomial and any polynomial. C Multiply: 5x 2 (x 3  4x 2 + 3x  5) Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a) x + 3 and x + 5 Solution a) (x + 3)(x + 5) = = (x  x) + (x  5) + (3  x) + (3  5) = x 2 + 5x + 3x + 15 = x 2 + 8x Multiplication of Polynomials c Multiply two binomials. DMultiply the following. (continued) Slide 12Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. x(x + 5) + 3(x + 5)

EXAMPLE Solution b) (3x  2)(x  1) = = (3x  x)  (3x  (  1)) +(  2  x)+ 2(  1) = = 3x 2  5x Multiplication of Polynomials c Multiply two binomials. DMultiplying binomials. Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 3x(x – 1)  2(x  1) 3x 2  3x  2x + 2

4.5 Multiplication of Polynomials Product of Two Polynomials Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. To multiply two polynomials P and Q, select one of the polynomials, say P. Then multiply each term of P by every term of Q and collect like terms.

EXAMPLE Solution 5x 3 + x 2 + 4x x 2 + 3x 15x 4 + 3x x 2 5x 5 + x 4 + 4x 3 5x x 4 + 7x x 2 Multiplying the top row by 3x Multiplying the top row by x 2 Collecting like terms 4.5 Multiplication of Polynomials d Multiply any two polynomials. EMultiply: (5x 3 + x 2 + 4x)(x 2 + 3x) Slide 15Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution 2x 2  3x + 1  3x 2  4  8x x  4  6x 4 + 9x 3  3x 2  6x 4 + 9x 3  11x x  4 Multiplying by  4 Collecting like terms 4.5 Multiplication of Polynomials d Multiply any two polynomials. F Multiply: (  3x 2  4)(2x 2  3x + 1) Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Multiplying by  3x 2