Explanation Given information: The polynomial is 16 x 4 − 1 . Formula used: Steps to determine the factors for a polynomial: Step 1: First look for a common factor and take out the largest common factor, if there is any common factor. Step 2: Observe the number of terms, for two terms-find if there is a difference of squares, for three terms-find if the trinomial is a square and for four terms-find factors by grouping. Step 3: Factor the terms completely, there should not be any factor that can be factorized further. Step 4: Check the result by multiplying the factors. The factorization of a difference of square is, A 2 − B 2 = ( A + B ) ( A − B ) Calculation: Consider the provided polynomial 16 x 4 − 1 . The polynomial 16 x 4 − 1 has only two terms. Express the two terms as perfect squares: 16 x 4 = ( 4 x 2 ) 2 and 1 = ( 1 ) 2 As both the terms are perfect squares the polynomial 16 x 4 − 1 can be factorized by difference of squares. Factor the expression 16 x 4 − 1 using the difference of squares as follows: 16 x 4 − 1 = ( 4 x 2 + 1 ) ( 4 x 2 − 1 ) Factorize the last polynomial ( 4 x 2 − 1 ) further using the difference of squares as follows: Express the two terms as perfect squares: 4 x 2 = ( 2 x ) 2 and 1 = ( 1 ) 2 As both the terms are perfect squares the polynomial ( 4 x 2 − 1 ) can be factorized by difference of squares. Factor the expression ( 4 x 2 − 1 ) using the difference of squares as follows: ( 4 x 2 − 1 ) = ( 2 x + 1 ) ( 2 x − 1 ) None of the resultant factors can be factorized further

9th Edition

ISBN: 9780321997173

Author: Marvin L. Bittinger, Judith A. Beecher

Publisher: PEARSON

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Chapter 11, Problem 15RE

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The factorization of the polynomial 16 x 4 − 1 .

Solution Summary: The author calculates the factorization of the polynomial 16x4-1 using the difference of squares.

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