Explanation Given information: The polynomial equation is as follows, P ( x ) = 2 x 4 + 15 x 3 + 31 x 2 + 20 x + 4 Theorems used: The Rational zeros theorem states that if the polynomial P ( x ) = a n x n + a n − 1 x n − 1 + ................ + a 0 has integer coefficients then every rational zero of P ( x ) is of the form, p q Here, p is a factor of constant coefficient a 0 and q is the factor of leading coefficient a n . The quadratic formula states that the quadratic equation R ( x ) = a x 2 + b x + c has its zeros calculated as follows, x = − b ± b 2 − 4 a c 2 a Calculation: By the use of Rational zeros theorem, the factors of a 0 are 1 , 2 , 4 and the factors of a n are 1 and 2 . Therefore, the possible rational zeros of the polynomial are ± 1 1 , ± 2 1 , ± 4 1 , ± 1 2 , ± 2 2 , ± 4 2 . By deleting the repeated possibilities, the set of all possible rational zeros are ± 1 , ± 2 , ± 4, ± 1 2 . Since every coefficient in the given polynomial is positive, there can’t be any positive number which makes the value of the polynomial zero. Use synthetic division to divide the polynomial P ( x ) by x − ( − 1 ) as shown below. − 1 2 15 31 20 4 − 2 − 13 − 18 − 2 _ 2 13 18 2 2 Since the remainder is not equal to 0 , therefore x = − 1 is not a zero of the polynomial. Use synthetic division to divide the polynomial P ( x ) by x − ( − 2 ) as shown below. − 2 2 15 31 20 4 − 4 − 22 − 18 − 4 _ 2 11 9 2 0 Since the remainder is 0 , therefore x = − 2 is a zero of the polynomial and hence the polynomial P ( x ) can be written in the form, P ( x ) = ( x − x 0 ) Q ( x ) Here, x 0 is the zero of the polynomial P ( x ) and Q ( x ) is the polynomial derived from the quotient row of the synthetic division

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 3.4, Problem 82E

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All the rational and irrational zeros of the polynomial P(x)=2x4+15x3+31x2+20x+4.

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Chapter 3.4, Problem 81E

Chapter 3.4, Problem 83E

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All the rational and irrational zeros of the polynomial P ( x ) = 2 x 4 + 15 x 3 + 31 x 2 + 20 x + 4 .

Solution Summary: The author states that the rational and irrational zeros of the polynomial P(x) are -2,-12 and the

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To calculate:

All the rational and irrational zeros of the polynomial P(x)=2x4+15x3+31x2+20x+4.

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Chapter 3 Solutions

All the rational and irrational zeros of the polynomial P ( x ) = 2 x 4 + 15 x 3 + 31 x 2 + 20 x + 4 .

Solution Summary: The author states that the rational and irrational zeros of the polynomial P(x) are -2,-12 and the

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To calculate: All the rational and irrational zeros of the polynomial P(x)=2x4+15x3+31x2+20x+4.

Want to see the full answer?

Chapter 3 Solutions

To calculate:

All the rational and irrational zeros of the polynomial P(x)=2x4+15x3+31x2+20x+4.