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Multiplying and Dividing Polynomials Chapter 5 Sections 5.4-5.7

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Copyright © Cengage Learning. All rights reserved. Multiplication of Monomials 5.4

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In the expression 2 4, the number 2 is called the base, and 4 is called the exponent. The expression may also be called the fourth power of 2. In x 3, the letter x is called the base and 3 is called the exponent. Multiplication of Monomials

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a. x 3 x 2 = (x x x)(x x) b. a 4 a 2 = (a a a a)(a a) c. c 5 c 3 = (c c c c c)(c c c) = x 5 = a 6 = c 8 Rule 1 for Exponents: Multiplying Powers a m a n = a m+n That is, to multiply powers with the same base, add the exponents.

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To multiply two monomials, multiply their numerical coefficients and combine their variable factors according to Rule 1 for exponents. Multiply: (2x 3 )(5x 4 ) (2x 3 )(5x 4 ) = 2 5 x 3 x 4 = 10x 3 + 4 = 10x 7 Multiplication of Monomials Add the exponents.

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Rule 2 for Exponents: Raising a Power to a Power (a m ) n = a mn That is, to raise a power to a power, multiply the exponents. (x 3 ) 5 = x 3 5 = x 15

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Rule 3 for Exponents: Raising a Product to a Power (ab) m = a m b m That is, to raise a product to a power, raise each factor to that same power. Example 6 (xy) 3 (xy) 3 = x 3 y 3 Example 7 (-2h 3 k 6 m 2 ) 5 = (-2) 5 (h 3 ) 5 (k 6 ) 5 (m 2 ) 5 = -32h 15 k 30 m 10 Multiplication of Monomials

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A special note about the meaning of – x 2 is needed here. Note that x is squared, not – x. That is, – x 2 = – (x x) If – x is squared, we have (– x) 2 = (– x)(– x) = x 2. Multiplication of Monomials (-3x 4 ) 4 = (-3) 4 (x 4 ) 4 = 81 x 16 -(3xy) 2 = - 9 x 2 y 2

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Copyright © Cengage Learning. All rights reserved. Multiplication of Polynomials 5.5

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To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial, and then add the products as shown in the following example. Multiplication of Polynomials Multiply: 3a(a 2 – 2a + 1). 3a(a 2 – 2a + 1) = 3a(a 2 ) + 3a(–2a) + 3a(1) = 3a 3 – 6a 2 + 3a

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Multiplying Two Polynomials To multiply one polynomial by another simply distribute both the terms of the first to the terms of the second, then combine like terms. (5x – 3) (2x + 4) = (5x)(2x) + (5x)(+4) + (-3)(2x) + (-3)(+4) = 10x 2 + 20x + (-6x) + (-12) = 10 x 2 + 14x -12 Some people think of this as FOIL meaning multiply First terms, Outer terms, Inner terms, then Last terms. Then combine all like terms.

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FOIL Some people think of this as FOIL meaning multiply First terms, Outer terms, Inner terms, then Last terms. Then combine all like terms. (2x + 5 ) ( x + 3) First terms: (2x)(x) Outer terms: (2x)(3) Inner terms: (5)(x) Last terms (5)(3) 2x 2 + 6x + 5x + 15 = 2x 2 + 11x + 15

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Copyright © Cengage Learning. All rights reserved. Division by a Monomial 5.6

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To divide a monomial by a monomial, first write the quotient in fraction form. 1.Write the problem in fraction form. 2.Reduce to lowest terms by dividing both the numerator and the denominator by their common factors. 3.Subtract any exponents of like terms. 4.Write your final answers. Division by a Monomial

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Examples 6a 2 : 2a = 6a 2 = 3 a 2-1 = 3a 2a 6a 2 b c 3 = 3 a 2-1 b 1-2 c 3-1 = 3 a b -1 c 2 10 a b 2 c 5 5 Note: negative exponents should be placed in the denominator so final answer would be 3 a c 2 5 b

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Divide: Example 4 Divide each term of the polynomial by 3x.

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Copyright © Cengage Learning. All rights reserved. Division by a Polynomial 5.7

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Dividing a polynomial by a polynomial of more than one term is very similar to long division in arithmetic. We use the same names, as shown below. Division by a Polynomial

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Step: 1. Divide the first term of the dividend, 2x 2, by the first term of the divisor, x. Write the result, 2x, above the dividend in line 1. Example 1 cont’d

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2. Multiply the divisor, x + 3, by 2x and write the result, 2x 2 + 6x, in line 3 as shown. 3. Subtract this result, 2x 2 + 6x, from the first two terms of the dividend, leaving –5x. Bring down the last term of the dividend, –14, as shown in line 4. 4. Divide the first term in line 4, –5x, by the first term of the divisor, x. Write the result, –5, to the right of 2x above the dividend in line 1. Example 1 cont’d

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5. Multiply the divisor, x + 3, by –5 and write the result, –5x – 15, in line 5 as shown. 6. Subtract line 5 from line 4, leaving 1 as the remainder. Since the remainder, 1, is of a lower degree than the divisor, x + 3, the problem is finished. The quotient is 2x – 5 with remainder 1. Example 1 cont’d

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As a check, you may use the relationship dividend = divisor quotient + remainder. Division by a Polynomial

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The dividend should always be arranged in decreasing order of degree. Any missing powers of x should be filled in by using zeros as coefficients. For example, x 3 – 1 = x 3 + 0x 2 + 0x – 1. Division by a Polynomial

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Resources for Practice Pg. 234 29, 31, 33, 37-38 Pg. 236 1, 13, 27, 40, 50 Pg. 238 1-2 Video resource: http://www.youtube.com/watch?v=Il5JgYl4R2 4 http://www.youtube.com/watch?v=Il5JgYl4R2 4

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