Slide 7- 1 Copyright © 2012 Pearson Education, Inc.

5.2 Multiplication of Polynomials ■ Multiplying Monomials ■ Multiplying Monomials and Binomials ■ Multiply Any Two Polynomials ■ The Product of Two Binomials: FOIL ■ Squares of Binomials ■ Products of Sums and Differences ■ Function Notation

Slide 5- 3 Copyright © 2012 Pearson Education, Inc. Multiply: a) (6x)(7x) b) (5a)(  a) c) (  8x 6 )(3x 4 ) Solution a) (6x)(7x) = (6  7) (x  x) = 42x 2 b) (5a)(  a) = (5a)(  1a) = (5)(  1)(a  a) =  5a 2 c) (  8x 6 )(3x 4 ) = (  8  3) (x 6  x 4 ) =  24x =  24x 10 Example

Slide 5- 4 Copyright © 2012 Pearson Education, Inc. Multiply: a) x and x + 7 b) 6x(x 2  4x + 5) Solution a) x(x + 7) = x  x + x  7 = x 2 + 7x b) 6x(x 2  4x + 5) = (6x)(x 2 )  (6x)(4x) + (6x)(5) = 6x 3  24x x Example

Slide 5- 5 Copyright © 2012 Pearson Education, Inc. Multiply each of the following. a) x + 3 and x + 5b) 3x  2 and x  1 Solution a) ( x + 3)(x + 5) = (x + 3)x + (x + 3)5 = x(x + 3) + 5(x + 3) = x  x + x   x + 5  3 = x 2 + 3x + 5x + 15 = x 2 + 8x + 15 Example

Slide 5- 6 Copyright © 2012 Pearson Education, Inc. Solution b) (3x  2)(x  1) = (3x  2)x  (3x  2)1 = x(3x  2)  1(3x  2) = x  3x  x  2  1  3x  1(  2) = 3x 2  2x  3x + 2 = 3x 2  5x + 2 continued

Slide 5- 7 Copyright © 2012 Pearson Education, Inc. The Product of Two Polynomials The product of two polynomials P and Q, is found by multiplying each term of P by every term of Q and combining like terms.

Slide 5- 8 Copyright © 2012 Pearson Education, Inc. Multiply: (5x 3 + x 2 + 4x)(x 2 + 3x). 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 Example

Slide 5- 9 Copyright © 2012 Pearson Education, Inc. The FOIL Method To multiply two binomials, A + B and C + D, multiply the First terms AC, the Outer terms AD, the Inner terms BC, and then the Last terms BD. Then combine like terms, if possible. (A + B)(C + D) = AC + AD + BC + BD Multiply First terms: AC. Multiply Outer terms: AD. Multiply Inner terms: BC Multiply Last terms: BD ↓ FOIL (A + B)(C + D) O I F L

Slide Copyright © 2012 Pearson Education, Inc. Multiply: (x + 4)(x 2 + 3). Solution F O I L (x + 4)(x 2 + 3) = x 3 + 3x + 4x = x 3 + 4x 2 + 3x + 12 Example O I F L The terms are rearranged in descending order for the final answer.

Slide Copyright © 2012 Pearson Education, Inc. Multiply. a) (x + 8)(x + 5)b) (y + 4) (y  3) c) (5t 3 + 4t)(2t 2  1)d) (4  3x)(8  5x 3 ) Solution a) (x + 8)(x + 5)= x 2 + 5x + 8x + 40 = x x + 40 b) (y + 4) (y  3)= y 2  3y + 4y  12 = y 2 + y  12 Example

Slide Copyright © 2012 Pearson Education, Inc. Solution c) (5t 3 + 4t)(2t 2  1) = 10t 5  5t 3 + 8t 3  4t = 10t 5 + 3t 3  4t d) (4  3x)(8  5x 3 ) = 32  20x 3  24x + 15x 4 = 32  24x  20x x 4 Example continued In general, if the original binomials are written in ascending order, the answer is also written that way.

Slide Copyright © 2012 Pearson Education, Inc. Squaring a Binomial (A + B) 2 = A 2 + 2AB + B 2 ; (A – B) 2 = A 2 – 2AB + B 2 The square of a binomial is the square of the first term, plus twice the product of the two terms, plus the square of the last term. Trinomials that can be written in the form A 2 + 2AB + B 2 or A 2 – 2AB + B 2 are called perfect-square trinomials.

Slide Copyright © 2012 Pearson Education, Inc. Multiply. a) (x + 8) 2 b) (y  7) 2 c) (4x  3x 5 ) 2 Solution (A + B) 2 = A 2 +2  A  B + B 2 a) (x + 8) 2 = x  x  = x x + 64 Example

Slide Copyright © 2012 Pearson Education, Inc. Example continued Solution (A – B) 2 = A 2  2AB + B 2 b) (y  7) 2 = y 2  2  y  = y 2  14y + 49 c) (4x  3x 5 ) 2 = (4x) 2  2  4x  3x 5 + (3x 5 ) 2 = 16x 2  24x 6 + 9x 10

Slide Copyright © 2012 Pearson Education, Inc. The Product of a Sum and Difference The product of the sum and difference of the same two terms is the square of the first term minus the square of the second term. (A + B)(A – B) = A 2 – B 2. This is called a difference of squares.

Slide Copyright © 2012 Pearson Education, Inc. Multiply. a) (x + 8)(x  8) b) (6 + 5w) (6  5w) c) (4t 3  3)(4t 3 + 3) Solution (A + B)(A  B) = A 2  B 2 a) (x + 8)(x  8)= x 2  8 2 = x 2  64 Example

Slide Copyright © 2012 Pearson Education, Inc. Example continued Solution b) (6 + 5w) (6  5w) = 6 2  (5w) 2 = 36  25w 2 c) (4t 3  3)(4t 3 + 3) = (4t 3 ) 2  3 2 = 16t 6  9

Slide Copyright © 2012 Pearson Education, Inc. Function Notation Example Given f(x) = x 2 – 6x + 7, find and simplify each of the following. a. f(a) + 4b) f(a + 3) Solution a. To find f(a) + 4, we replace x with a to find f(a). Then we add 4 to the result. f(a) + 4 = a 2 – 6a = a 2 – 6a + 11

Slide Copyright © 2012 Pearson Education, Inc. continued f(x) = x 2 – 6x + 7 b) f(a + 3) f(a + 3) = (a + 3) 2 – 6(a + 3) + 7 = a 2 + 6a + 9 – 6a – = a 2 – 2.