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 possible 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 5 and q is the factor of 3 . The possibilities of p q are, Possibilities for p Possibilities for q : ± 1 , ± 5 ± 1 , ± 3 Thus, the possibilities for the rational zeros of the function f ( x ) = 3 x 3 − x 2 − 15 x + 5 is 1 , − 1 , 5 , − 5 , 1 3 , − 1 3 , 5 3 , − 5 3 . Try for the above possibilities, if f ( x ) = 0 . Here, 1 3 satisfies the condition f ( 1 3 ) = 0 , hence 1 3 is a zero of f ( x ) . As 1 3 is one of the zero, it is possible to write the polynomial as, f ( x ) = ( x − 1 3 ) ⋅ Q ( x ) Synthetic division is used to find the value of Q ( x ) b) To determine The factor the polynomial into linear factors.

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Author: BITTINGER

Publisher: PEARSON

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Chapter 4.4, Problem 57E

To determine

The possible rational zeros and the other zeros of the function f(x)=3x3−x2−15x+5.

To determine

The factor the polynomial into linear factors.

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Chapter 4.4, Problem 56E

Chapter 4.4, Problem 58E

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The possible rational zeros and the other zeros of the function f ( x ) = 3 x 3 − x 2 − 15 x + 5 .

Solution Summary: The author explains the rational zeros of the function f(x): p and q are relatively prime.