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 16 and q is the factor of 1 . The possibilities of p q are, Possibilities for p Possibilities for q : ± 1 , ± 2 , ± 4 , ± 8 , ± 16 ± 1 Thus, the possibilities for the rational zeros of the function f ( x ) = x 3 + 4 x 2 + 4 x + 16 is 1 , − 1 , 2 , − 2 , 4 , − 4 , 8 , − 8 , 16 , − 16 . Try for the above possibilities, if f ( x ) = 0 . Here, –3 satisfies the condition f ( − 4 ) = 0 . So, ‒4 is a zero of f ( x ) . As –3 is one of the zero, it is possible to write the polynomial as, f ( x ) = ( x + 4 ) ⋅ Q ( x ) Synthetic division is used to find the value of Q ( x ) b) To determine The factor the polynomial into linear factors.

6th Edition

ISBN: 2810000027802

Author: BITTINGER

Publisher: PEARSON

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Chapter 4, Problem 21T

To determine

The possible rational zeros and the other zeros of the function f(x)=x3+4x2+4x+16.

To determine

The factor the polynomial into linear factors.

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Chapter 4, Problem 20T

Chapter 4, Problem 22T

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The possible rational zeros and the other zeros of the function f ( x ) = x 3 + 4 x 2 + 4 x + 16 .

Solution Summary: The author explains the possible rational zeros of the function f(x) and the rational number denoted by pq.