Explanation Steps used: 1. Descartes’ Rule of Signs: To find the number of possible positive, negative, and non-real zeros. 2. Rational Zero Theorem: To list the possible rational zeros. 3. Synthetic Division: To find a zero. If there are more than one zeros, we need to repeat the previous steps. Given: P ( x ) = x 4 + 4 x 3 + 6 x 2 + 4 x + 1 . Calculation: Consider, P ( x ) = x 4 + 4 x 3 + 6 x 2 + 4 x + 1 . (1) Using Descartes’ Rule of Signs, There are four possible combinations of positive negative, and non-real zeros. The possible combinations of positive zeros are calculated by finding the number of changes in variation of sign in P ( x ) . Consider P ( x ) , P ( x ) = x 4 + 4 x 3 + 6 x 2 + 4 x + 1 Which shows that there are no possible combinations of positive zeros. The possible combinations of negative zeros are calculated by finding the number of changes in variation of sign in P ( − x ) . Consider P ( − x ) , P ( − x ) = ( − x ) 4 + 4 ( − x ) 3 + 6 ( − x ) 2 + 4 ( − x ) + 1 = x 4 − 4 x 3 + 6 x 2 − 4 x + 1 P ( − x ) = x 4 − ⎵ 1 4 x 3 + ⎵ 2 6 x 2 − ⎵ 3 4 x + ⎵ 4 1 Which shows that there are 4 possible combinations of negative zeros. Thus, the possible combinations of positive, negative and non-real zeros are given as, Number of Positive Zeros Number of Negative Zeros Number of Non-real Zeros 0 4 0 0 3 1 0 2 2 0 1 3 (2) Using Rational Zero Theorem we can find the possible rational zeros which have the form factor of constant factor of leading coefficient . Here, The factor of a constant term in P ( x ) is 1 and the factor of the leading coefficient is 1 factor of constant factor of leading coefficient = 1 1 , therefore the possible rational zeros are, ± 1. (3) Using the Synthetic division we can check for the possibility of each number whether it is a zero. Consider x = − 1 , − 1 1 4 6 4 1 0 − 1 − 3 − 3 − 1 1 3 3 1 0 The reminder is zero which shows that x = − 1 is one zero of P ( x )

12th Edition

ISBN: 9781305652231

Author: R. David Gustafson, Jeff Hughes

Publisher: Cengage Learning

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Chapter 4.5, Problem 30E

To determine

To find:

The all rational zeros of P(x)=x4+4x3+6x2+4x+1.

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Chapter 4.5, Problem 29E

Chapter 4.5, Problem 31E

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