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 possible rational zeros of the given function x 3 − 1 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 1 and q is the factor of 1. The possibilities of p q are, Possibilities for p Possibilities for q : ± 1 ± 1 Thus, the possibilities for the rational zeros of the function x 3 − 1 are ± 1 . Try for the above possibilities, if x 3 − 1 = 0 . Note that, 1 satisfies the condition x 3 − 1 = 0 , hence 1 is a zero. As 1 is one of the zero, it is possible to write the polynomial as, x 3 − 1 = ( x − 1 ) ⋅ Q ( x ) . Synthetic division is used to find the value of Q ( x ) . i 1 0 0 − 1 1 1 1 _ 1 1 1 0 Thus, the value of Q ( x ) = x 2 + x + 1

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ISBN: 9780135962312

Author: BITTINGER

Publisher: PEARSON

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Chapter 4, Problem 90RE

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To express: The polynomial x3−1 as a product of linear factors.

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Chapter 4, Problem 89RE

Chapter 4, Problem 91RE

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To express: The polynomial x 3 − 1 as a product of linear factors.

Solution Summary: The author describes the rational zeros of the polynomial x3-1 as a product of linear factors.

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