a) Explanation Theorem used: “ The rational zeros theorem: Let P ( x ) = a n x n + a n − 1 x n − 1 + … + a 1 x + a 0 , where all the coefficients are integers. Consider a rational number denoted by p q , where p and q are relatively prime. If p q is a zero of P ( x ) , then p is a factor of a 0 and q is the factor of a n . Calculation: Obtain the rational zeros of the given function as follows. Since the given polynomial is of degree 3, there are at most 3 distinct zeros. From the rational zeros theorem, p q to be a zero of P ( x ) , p should be a factor of a 0 and q is the factor of a n . That is, p should be a factor of 8 and q is the factor of 1. The possibilities of p q are, Possibilities for p Possibilities for q : ± 1 , ± 2 , ± 4 , ± 8 ± 1 Thus, the possibilities for the rational zeros of the function f ( x ) = x 3 − 8 is 1 , − 1 , 2 , − 2 , 4 , − 4 , 8 , − 8 . Try for the above possibilities, if f ( x ) = 0 . Here, 2 satisfies the condition f ( 2 ) = 0 , hence 2 is a zero of f ( x ) . As 2 is one of the zero, it is possible to write the polynomial as, f ( x ) = ( x − 2 ) ⋅ Q ( x ) Synthetic division is used to find the value of Q ( x ) b) To determine The factor the polynomial into linear factors.

College Algebra: Graphs and Models (6th Edition)

6th Edition

ISBN: 9780134179032

Author: Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen, Judith A. Penna

Publisher: PEARSON

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Chapter 4.4, Problem 70E

To determine

The rational zeros and the other zeros of the function f(x)=x3−8.

To determine

The factor the polynomial into linear factors.

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Chapter 4.4, Problem 69E

Chapter 4.4, Problem 71E

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The rational zeros and the other zeros of the function f ( x ) = x 3 − 8 .

Solution Summary: The author explains the rational zeros of the function f(x)=x 3-8. Since the given polynomial is of degree 3, there are at most 3 distinct zero