How do you factor ${(1 + x - x^{2})}^{2}$ $(1+x-x^2)^2$ ?

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How do you factor ${(1 + x - x^{2})}^{2}$ $(1+x-x^2)^2$ ?

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Answer 1 · 2016-08-28T18:09:11

${(1 + x - x^{2})}^{2} = {(x - \frac{1}{2} - \sqrt{\frac{5}{2}})}^{2} {(x - \frac{1}{2} + \frac{\sqrt{5}}{2})}^{2}$ $(1+x-x^2)^2=(x-1/2-sqrt(5/2))^2(x-1/2+sqrt(5)/2)^2$

Explanation:

${(1 + x - x^{2})}^{2}$ $(1+x-x^2)^2$

$= {(x^{2} - x - 1)}^{2}$ $=(x^2-x-1)^2$

$= {({(x - \frac{1}{2})}^{2} - \frac{5}{4})}^{2}$ $=((x-1/2)^2-5/4)^2$

$= {⎛ ⎝ {(x - \frac{1}{2})}^{2} - {(\frac{\sqrt{5}}{2})}^{2} ⎞ ⎠}^{2}$ $=((x-1/2)^2-(sqrt(5)/2)^2)^2$

$= {(x - \frac{1}{2} - \sqrt{\frac{5}{2}})}^{2} {(x - \frac{1}{2} + \frac{\sqrt{5}}{2})}^{2}$ $=(x-1/2-sqrt(5/2))^2(x-1/2+sqrt(5)/2)^2$

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How do you factor ${(1 + x - x^{2})}^{2}$ $(1+x-x^2)^2$ ?

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How do you factor ${(1 + x - x^{2})}^{2}$ $(1+x-x^2)^2$ ?