Explanation Procedure: The Rational Zeros Theorem states that, if the polynomial P x = a n x n + a n - 1 x n - 1 + … + a 1 x + a 0 has integer coefficients (where a n ≠ 0 and a 0 ≠ 0 ), then every rational zero of P is of the form p q , where p and q are integers and p is a factor of the constant term a 0 and q is a factor of the leading coefficient a n . To find the rational zeros of a polynomial use the following steps: 1. List all possible rational zeros, using the Rational Zeros Theorem. 2. Use synthetic division to evaluate the polynomial at each of the candidates for the rational zeros that you found in step 1. When the remainder is 0, note the quotient you have obtained. 3. Repeat steps 1 and 2 for the quotient. Stop when you reach a quotient that is quadratic or factors easily, and use the quadratic formula or factor to find the remaining zeros. The factor theorem states that, if c is a zero of a polynomial function P ( x ) , then ( x - c ) is a factor of P ( x ) . Formula: The zeros of the quadratic equation a x 2 + b x + c = 0 , where a ≠ 0 , can be obtained by the Quadratic Formula: x = - b ± b 2 - 4 a c 2 a Calculation: The given polynomial is P x = x 4 - 6 x 3 + 4 x 2 + 15 x + 4 . By the Rational Zeros Theorem, the possible rational zeros of P x are of the form: p q = f a c t o r o f c o n s t a n t t e r m f a c t o r o f l e a d i n g c o e f f i c i e n t So first find the factors of the constant term and the leading coefficient. Since the leading coefficient is 1, any rational zero must be a factor of the constant term. In the given polynomial, the constant term is 4 and its factors are ± 1 , ± 2 , ± 4 . So the possible rational zeros of P ( x ) are ± 1 , ± 2 , ± 4 . First let’s check whether 1 is a zero of P x by using synthetic division. 1 1 − 6 4 15 4 1 − 5 − 1 14 _ 1 − 5 − 1 14 18 Here the remainder is not 0 , so 1 is not a zero. Next let’s check whether - 1 is a zero. − 1 1 − 6 4 15 4 − 1 7 − 11 − 4 _ 1 − 7 11 4 0 Here the remainder is 0 , so - 1 is a zero. From the last line of the synthetic division we can see that the quotient is x 3 - 7 x 2 + 11 x + 4 . So from synthetic division P ( x ) factors as P x = x + 1 ( x 3 - 7 x 2 + 11 x + 4 ) …(1) We continue by factoring the quotient x 3 - 7 x 2 + 11 x + 4 . Since the leading coefficient is 1, any rational zero must be a factor of the constant term. Here the constant term is 4 and its factors are ± 1 , ± 2 , ± 4

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ISBN: 8220100548512

Author: Watson

Publisher: CENGAGE L

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Chapter 3.4, Problem 47E

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All real zeros of the polynomial Px=x4-6x3+4x2+15x+4.

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Chapter 3.4, Problem 46E

Chapter 3.4, Problem 48E

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