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 4, there are at most 4 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 2 and q is the factor of 3. The possibilities of p q are, Possibilities for p Possibilities for q : ± 1 , ± 2 ± 1 , ± 3 Thus, the possibilities for the rational zeros of the function f ( x ) = 3 x 4 − 4 x 3 + x 2 + 6 x − 2 are 1 , − 1 , 2 , − 2 , 1 3 , − 1 3 , 2 3 , − 2 3 . 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 3 − 4 1 6 − 2 − 3 7 − 8 2 _ 3 − 7 8 − 2 0 Thus, the value of Q ( x ) = ( 3 x 4 − 7 x 2 + 8 x − 2 ) b) To determine The factor of the polynomial into linear factors.

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ISBN: 9780135962312

Author: BITTINGER

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Chapter 4.4, Problem 64E

To determine

The possible rational zeros and the other zeros of the function f(x)=3x4−4x3+x2+6x−2.

To determine

The factor of the polynomial into linear factors.

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Chapter 4.4, Problem 63E

Chapter 4.4, Problem 65E

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The possible rational zeros and the other zeros of the function f ( x ) = 3 x 4 − 4 x 3 + x 2 + 6 x − 2 .

Solution Summary: The author explains the rational zeros of the function f(x): p and q are relatively prime. Since the given polynomial is of degree 4, there are at most 4 distinct zero