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 4 and q is the factor of 5. The possibilities of p q are, Possibilities for p Possibilities for q : ± 1 , ± 2 , ± 4 ± 1 , ± 5 Thus, the possibilities for the rational zeros of the function f ( x ) = 5 x 4 − 4 x 3 + 19 x 2 − 16 x − 4 are 1 , − 1 , 2 , − 2 , 4 , − 4 , 1 5 , − 1 5 , 2 5 , − 2 5 , 4 5 , − 4 5 . Try for the above possibilities, if f ( x ) = 0 . Here, 1 satisfies the condition f ( 1 ) = 0 , hence 1 is a zero of f ( x ) . As 1 is one of the zero, it is possible to write the polynomial as, f ( x ) = ( x − 1 ) ⋅ Q ( x ) Synthetic division is used to find the value of Q ( x ) . 1 5 − 4 19 − 16 − 4 5 1 b) To determine The factor the polynomial into linear factors.

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Chapter 4.4, Problem 63E

To determine

The possible rational zeros and the other zeros of the function f(x)=5x4−4x3+19x2−16x−4.

To determine

The factor the polynomial into linear factors.

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Chapter 4.4, Problem 62E

Chapter 4.4, Problem 64E

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The possible rational zeros and the other zeros of the function f ( x ) = 5 x 4 − 4 x 3 + 19 x 2 − 16 x − 4 .

Solution Summary: The author explains the rational zeros of the polynomial function f(x) and the other zero.