Explanation Given Information: The provided polynomial is 18 x 3 − 9 x 2 − 8 x + 4 . Formula used: Rational Zeros Theorem: If a polynomial ( a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a o ) has integer coefficient, then every rational zero is of the form x = p q , where p is a factor of a o , and q is a factor of a n . Calculation: Consider the polynomial, p ( x ) = 18 x 3 − 9 x 2 − 8 x + 4 Factor of constant term ( 4 ) of polynomial 18 x 3 − 9 x 2 − 8 x + 4 will be ± 1 , ± 2 and ± 4 . Factor of leading coefficient ( 18 ) will be ± 1 , ± 2, ± 3 , ± 6 and ± 18 . Now apply, Rational Zeros Theorem for polynomial p ( x ) to find zeros. The possible rational zeros are the factor of the constant term divided by the factors of the leading coefficient that is, ± 1 , ± 1 2 , ± , 1 6 ± , 1 3 , ± 1 9 , ± 2 9 , ± 4 9 , ± 2, ± 4 and ± 1 18 The possible rational factor of polynomial 18 x 3 − 9 x 2 − 8 x + 4 is ± 1 , ± 1 2 , ± , 1 6 ± , 1 3 , ± 2 3 , ± 2, ± 3 and ± 6 Now, apply synthetic devisor method for x = 1 2 . 1 2 18 − 9 − 8 4 9 0 − 4 18 0 − 8 0 The factor of polynomial 18 x 3 − 9 x 2 − 8 x + 4 are ( x − 1 2 ) ( 18 x 2 − 8 ) . Consider the quadratic equation, 18 x 2 − 8 Factor the quadratic equation 18 x 2 − 8 , ( 18 x 2 − 8 ) = 18 x 2 − 8 = 2 ( 9 x 2 − 4 ) = 2 ( 3 x + 2 ) ( 3 x − 2 ) The factor of polynomial 18 x 3 − 9 x 2 − 8 x + 4 are 2 ( x − 1 2 ) ( 3 x + 2 ) ( 3 x − 2 ) . To find zeros set first term of the factor 2 ( x − 1 2 ) ( 3 x + 2 ) ( 3 x − 2 ) equal to zero. ( x − 1 2 ) = 0 x = 1 2 To find zeros set second term factor 2 ( x − 1 2 ) ( 3 x + 2 ) ( 3 x − 2 ) equal to zero

Calculus: An Applied Approach (MindTap Course List)

10th Edition

ISBN: 9781305860919

Author: Ron Larson

Publisher: Cengage Learning

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Chapter A4, Problem 74E

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Chapter A4, Problem 73E

Chapter A4, Problem 75E

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Chapter A4 Solutions

Calculus: An Applied Approach (MindTap Course List)

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The factor the polynomial 18 x 3 − 9 x 2 − 8 x + 4 .

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Chapter A4 Solutions