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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 12 Exponents and Polynomials

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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 12.5 Multiplying Polynomials

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Martin-Gay, Developmental Mathematics, 2e 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. If all of the polynomials are monomials, use the associative and commutative properties. If any of the polynomials are not monomials, use the distributive property before the associative and commutative properties. Then combine like terms. Multiplying Polynomials

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Martin-Gay, Developmental Mathematics, 2e 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To Multiply Two Polynomials Multiply each term of the first polynomial by each term of the second polynomial, and then combine like terms. Multiplying Polynomials

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Martin-Gay, Developmental Mathematics, 2e 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply. a. (3x 2 )( – 2x) = (3)( – 2)(x 2 · x)= – 6x 3 b. (4x 2 )(3x 2 – 2x + 5) = (4x 2 )(3x 2 ) – (4x 2 )(2x) + (4x 2 )(5) Apply the distributive property. = 12x 4 – 8x 3 + 20x 2 Multiply the monomials. c. (2x – 4)(7x + 5)= 2x(7x + 5) – 4(7x + 5) = 14x 2 + 10x – 28x – 20 = 14x 2 – 18x – 20 Example

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Martin-Gay, Developmental Mathematics, 2e 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply (3x + 4) 2. Remember that a 2 = a · a, so (3x + 4) 2 = (3x + 4)(3x + 4). (3x + 4) 2 = (3x + 4)(3x + 4)= (3x)(3x + 4) + 4(3x + 4) = 9x 2 + 12x + 12x + 16 = 9x 2 + 24x + 16 Example

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Martin-Gay, Developmental Mathematics, 2e 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 2 + 14 = a 4 – a 3 – 6a 2 + 7a + 14 Example

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Martin-Gay, Developmental Mathematics, 2e 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply (5x – 2z) 2. (5x – 2z) 2 = (5x – 2z)(5x – 2z)= (5x)(5x – 2z) – 2z(5x – 2z) = 25x 2 – 10xz – 10xz + 4z 2 = 25x 2 – 20xz + 4z 2 Example

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Martin-Gay, Developmental Mathematics, 2e 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply (2x 2 + x – 1)(x 2 + 3x + 4). (2x 2 + x – 1)(x 2 + 3x + 4) = (2x 2 )(x 2 + 3x + 4) + x(x 2 + 3x + 4) – 1(x 2 + 3x + 4) = 2x 4 + 6x 3 + 8x 2 + x 3 + 3x 2 + 4x – x 2 – 3x – 4 = 2x 4 + 7x 3 + 10x 2 + x – 4 Example

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Martin-Gay, Developmental Mathematics, 2e 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Another convenient method for multiplying polynomials is to multiply vertically, similar to the way we multiply real numbers. In this case, as each term of one polynomial is multiplied by a term of the other polynomial, the partial products are aligned so that like terms are together. This can make it easier to find and combine like terms. Multiplying Polynomials

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