Explanation Given Information: The provided polynomial is 6 x 3 − 11 x 2 − 19 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 ) = 6 x 3 − 11 x 2 − 19 x − 6 Factor of constant term ( − 6 ) of polynomial 6 x 3 − 11 x 2 − 19 x − 6 will be ± 1 , ± 2 ± 3 and ± 6 . Factor of leading coefficient ( 6 ) will be ± 1 , ± 2 ± 3 and ± 6 . 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 , ± 2 3 , ± 2, ± 3 and ± 6 The possible rational factor of polynomial 6 x 3 − 11 x 2 − 19 x − 6 is ± 1 , ± 1 2 , ± , 1 6 ± , 1 3 , ± 2 3 , ± 2, ± 3 and ± 6 Now, apply synthetic devisor method for x = 3 . 3 6 − 11 − 19 − 6 18 21 6 6 7 2 0 The factor of polynomial 6 x 3 − 11 x 2 − 19 x − 6 are ( x − 3 ) ( 6 x 2 + 7 x + 2 ) . Consider the quadratic equation, 6 x 2 + 7 x + 2 Factor the quadratic equation 6 x 2 + 7 x + 2 , ( 6 x 2 + 7 x + 2 ) = 6 x 2 + 4 x + 3 x + 2 = 2 x ( 3 x + 2 ) + 1 ( 3 x + 2 ) = ( 3 x + 2 ) ( 2 x + 1 ) The factor of polynomial 6 x 3 − 11 x 2 − 19 x − 6 are ( x − 3 ) ( 3 x + 2 ) ( 2 x + 1 ) . To find zeros set first term of the factor ( x − 3 ) ( 3 x + 2 ) ( 2 x + 1 ) equal to zero. ( x − 3 ) = 0 x = 3 To find zeros set second term factor ( x − 3 ) ( 3 x + 2 ) ( 2 x + 1 ) equal to zero

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ISBN: 8220101426222

Author: Larson

Publisher: CENGAGE L

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Chapter A4, Problem 73E

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Chapter A4, Problem 72E

Chapter A4, Problem 74E

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Chapter A4 Solutions

EBK CALCULUS: AN APPLIED APPROACH

Ch. A4 - Prob. 1CP Ch. A4 - Prob. 2CP Ch. A4 - Prob. 3CP Ch. A4 - Prob. 4CP Ch. A4 - Prob. 1E Ch. A4 - Prob. 2E Ch. A4 - Prob. 3E Ch. A4 - Prob. 4E Ch. A4 - Prob. 5E Ch. A4 - Prob. 6E

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The factor the polynomial 6 x 3 − 11 x 2 − 19 x − 6 .

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Chapter A4 Solutions