Explanation Given Information: The provided polynomial is 3 x 3 + 4 x 2 − 13 x + 6 . 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 ) = 3 x 3 + 4 x 2 − 13 x + 6 Factor of constant term ( 6 ) of polynomial 3 x 3 + 4 x 2 − 13 x + 6 will be ± 1 , ± 2 ± 3 and ± 6 . Factor of leading coefficient ( 3 ) will be ± 1 , ± 2 and ± 3 . 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 3 , ± 2 3 , ± 2, ± 3 and ± 6 The possible rational factor of polynomial 3 x 3 + 4 x 2 − 13 x + 6 is ± 1 , ± 1 2 , ± 1 3 , ± 3 2 , ± 2, ± 3 and ± 6 . Now, apply synthetic devisor method for x = − 3 . − 3 3 4 − 13 6 − 9 15 − 6 3 − 5 2 0 The factor of polynomial 3 x 3 + 4 x 2 − 13 x + 6 are ( x + 3 ) ( 3 x 2 − 5 x + 2 ) . Consider the quadratic equation, 3 x 2 − 5 x + 2 Factor the quadratic equation 3 x 2 − 5 x + 2 , ( 3 x 2 − 5 x + 2 ) = 3 x 2 − 3 x − 2 x + 2 = 3 x ( x − 1 ) − 2 ( x − 1 ) = ( x − 1 ) ( 3 x − 2 ) The factor of polynomial 3 x 3 + 4 x 2 − 13 x + 6 are ( x − 1 ) ( x + 3 ) ( 3 x − 2 ) . To find zeros set first term of the factor ( x − 1 ) ( x + 3 ) ( 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 72E

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Chapter A4, Problem 71E

Chapter A4, Problem 73E

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Chapter A4 Solutions

Calculus: An Applied Approach (MindTap Course List)

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The factor the polynomial 3 x 3 + 4 x 2 − 13 x + 6 .

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