Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Chapter 4 Polynomials

1-2 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Special Products Products of Two Binomials Multiplying Sums and Differences of Two Terms Squaring Binomials Multiplications of Various Types 4.6

Copyright © 2014, 2010, and 2006 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

1-4 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example 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 O I F L The terms are rearranged in descending order for the final answer.

1-5 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example 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

1-6 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example 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 In general, if the original binomials are written in ascending order, the answer is also written that way.

Copyright © 2014, 2010, and 2006 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.

1-8 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example 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

1-9 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example 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

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. The Square of a Binomial 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. (A + B) 2 = A 2 + 2AB + B 2 ; (A – B) 2 = A 2 – 2AB + B 2 ; These are called perfect-square trinomials.* *Another name for these is trinomial squares.

1-11 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example 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

1-12 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Solution 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

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Multiplying Two Polynomials 1.Is the multiplication the product of a monomial and a polynomial? If so, multiply each term of the polynomial by the monomial. 2. Is the multiplication the product of two binomials? If so: a) Is the product of the sum and difference of the same two terms? If so, use the pattern (A + B)(A  B) = (A  B) 2. b) Is the product the square of a binomial? If so, use the pattern (A + B) 2 = A 2 + 2AB + B 2, or (A – B) 2 = A 2 – 2AB + B 2. c) If neither (a) nor (b) applies, use FOIL. 3.Is the multiplication the product of two polynomials other than those above? If so, multiply each term of one by every term of the other. Use columns if you wish.

1-14 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Multiply: a) (x + 5)(x  5)b) (w  7)(w + 4) c) (x + 9)(x + 9)d) 3x 2 (4x 2 + x  2) e) (p + 2)(p 2 + 3p  2)f) (2x + 1) 2 Solution a) (x + 5)(x  5)= x 2  25 b) (w  7)(w + 4) = w 2 + 4w  7w  28 = w 2  3w  28

1-15 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example c) (x + 9)(x + 9)= x x + 81 d) 3x 2 (4x 2 + x  2) = 12x 4 + 3x 3  6x 2 e) p 2 + 3p  2 p + 2 2p 2 + 6p  4 p 3 + 3p 2  2p p 3 + 5p 2 + 4p  4

1-16 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example f) (2x + 1) 2 = 4x 2 + 2(2x)(1) + 1 = 4x 2 + 4x + 1