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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 3.5 Multiplying Polynomials

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 22 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, Prealgebra & Introductory Algebra, 3ed 33 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, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply each of the following. 1) (3x 2 )( – 2x) = (3)( – 2)(x 2 · x) = – 6x 3 2) (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. 3) (2x – 4)(7x + 5)= 2x(7x + 5) – 4(7x + 5) = 14x 2 + 10x – 28x – 20 = 14x 2 – 18x – 20 Multiplying Polynomials Example

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 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 Multiplying Polynomials Example

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66 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 Multiplying Polynomials Example

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 Multiplying Polynomials Example

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 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 Multiplying Polynomials Example

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 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 Multiplying Polynomials Example

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 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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