71 R.6 Factoring Polynomials 132. Geometric Modeling Explain how the figures give geometric interpretation to the formula x2 + 2xy + y2 = 1x + y22. x x y y y y x x y y x Factor each polynomial over the set of rational number coefficients. 133. 49x2 - 1 25 134. 81y2 - 1 49 135. 25 9 x4 - 9y2 136. 121 25 y4 - 49x2 Concept Check Find all values of b or c that will make the polynomial a perfect square trinomial. 137. 4z2 + bz + 81 138. 9p2 + bp + 25 139. 100r2 - 60r + c 140. 49x2 + 70x + c Relating Concepts For individual or collaborative investigation (Exercises 141–146) The polynomial x6 - 1 can be considered either a difference of squares or a difference of cubes. Work Exercises 141–146 in order, to connect the results obtained when two different methods of factoring are used. 141. Factor x6 - 1 by first factoring as a difference of squares, and then factor further by using the patterns for a sum of cubes and a difference of cubes. 142. Factor x6 - 1 by first factoring as a difference of cubes, and then factor further by using the pattern for a difference of squares. 143. Compare the answers in Exercises 141 and 142. Based on these results, what is the factorization of x4 + x2 + 1? 144. The polynomial x4 + x2 + 1 cannot be factored using the methods described in this section. However, there is a technique that enables us to factor it, as shown here. Supply the reason why each step is valid. x4 + x2 + 1 = x4 + 2x2 + 1 - x2 = 1x4 + 2x2 + 12 - x2 = 1x2 + 122 - x2 = 1x2 + 1 - x21x2 + 1 + x2 = 1x2 - x + 121x2 + x + 12 145. How does the answer in Exercise 143 compare with the final line in Exercise 144? 146. Factor x8 + x4 + 1 using the technique outlined in Exercise 144.
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