Copyright 2013, 2009, 2005, 2002 Pearson, Education, Inc.

5.4 Multiplying Polynomials

Multiply. a. (3x 2 )( – 2x) = (3)( – 2)(x 2 · x)= – 6x 3 b. 4x(4x 3 + 8) c. Example

Multiply.

To multiply any two polynomials, use the distributive property and multiply each term of one polynomial by each term of the other polynomial. Then combine any like terms. Multiplying Two Polynomials

Multiply. a. b. (2x – 4)(7x + 5)= 2x(7x + 5) – 4(7x + 5) = 14x x – 28x – 20 = 14x 2 – 18x – 20 Example

Multiply (a + 2)(a 3 – 3a 2 + 7). (a + 2)(a 3 – 3a 2 + 7) = a(a 3 – 3a 2 + 7) + 2(a 3 – 3a 2 + 7) = a 4 – 3a 3 + 7a + 2a 3 – 6a = a 4 – a 3 – 6a 2 + 7a + 14 Example

Multiply (2y 2 + 5)(y 2 + 3y + 4) vertically. Example

The FOIL Method When multiplying 2 binomials, the distributive property can be easily remembered as the FOIL method. F – product of First terms O – product of Outside terms I – product of Inside terms L – product of Last terms

= y 2 – 8y – 48 Multiply (y – 12)(y + 4). (y – 12)(y + 4) Product of First terms is y 2 Product of Outside terms is 4y Product of Inside terms is – 12y Product of Last terms is – 48 (y – 12)(y + 4) = y 2 + 4y – 12y – 48 F O I L Example

Multiply (2x – 4)(7x + 5) (2x – 4)(7x + 5) = = 14x x – 28x – 20 F 2x(7x) F + 2x(5) O – 4(7x) I – 4(5) L O I L = 14x 2 – 18x – 20 Example

Multiply (3x – 7y)(7x + 2y) (3x – 7y)(7x + 2y)= (3x)(7x + 2y) – 7y(7x + 2y) = 21x 2 + 6xy – 49xy + 14y 2 = 21x 2 – 43xy + 14y 2 Example

A binomial squared is equal to the square of the first term plus or minus twice the product of both terms plus the square of the second term. (a + b) 2 = a 2 + 2ab + b 2 (a – b) 2 = a 2 – 2ab + b 2 Squaring a Binomial

Example Multiply. (x + 6) 2 F OI L (x + 6) 2 = x 2 + 6x + 6x + 36 = x x + 36 The inner and outer products are the same. = (x + 6)(x + 6)

Example a. (12a – 3) 2 = 144a 2 – 72a + 9 = (12a) 2 – 2(12a)(3) + (3) 2 b. (x + y) 2 = x 2 + 2xy + y 2 Multiply.

Product of the Sum and Difference of Two Terms The product of the sum and difference of two terms is the square of the first term minus the square of the second term. (a + b)(a – b) = a 2 – b 2 Multiplying the Sum and Difference of Two Terms

Example Multiply. = (2x)(2x) + (2x)(– 4) + (4)(2x) + (4)(– 4) F OI L (2x + 4)(2x – 4) = 4x 2 + (– 8x) + 8x + (–16) = 4x 2 – 16 The inner and outer products cancel.

Example Multiply. a. (5a + 3)(5a – 3) = 25a 2 – 9 = (5a) 2 – 3 2 b. (8c + 2d)(8c – 2d) = 64c 2 – 4d 2 = (8c) 2 – (2d) 2

Example Multiply

Example Multiply

Techniques of multiplying polynomials are often useful when evaluating polynomial functions at polynomial values. If f(x) = 2x 2 + 3x – 4, find f(a + 3). We replace the variable x with a + 3 in the polynomial function. f(a + 3) = 2(a + 3) 2 + 3(a + 3) – 4 = 2(a 2 + 6a + 9) + 3a + 9 – 4 = 2a a a + 9 – 4 = 2a a + 23 Evaluating Polynomials Example