Factorize:
x2+1x2−11x^2 + \dfrac{1}{x^2} - 11x2+x21−11
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Given,
⇒x2+1x2−11⇒(x2+1x2−2)−9⇒(x2+1x2−2×x×1x)−9⇒(x−1x)2−(3)2⇒(x−1x+3)(x−1x−3).\Rightarrow x^2 + \dfrac{1}{x^2} - 11 \\[1em] \Rightarrow \Big(x^2 + \dfrac{1}{x^2} - 2\Big) - 9 \\[1em] \Rightarrow \Big(x^2 + \dfrac{1}{x^2} - 2 \times x \times \dfrac{1}{x}\Big) - 9 \\[1em] \Rightarrow \Big(x - \dfrac{1}{x}\Big)^2 - (3)^2 \\[1em] \Rightarrow \Big(x - \dfrac{1}{x} + 3\Big) \Big(x - \dfrac{1}{x} - 3\Big).⇒x2+x21−11⇒(x2+x21−2)−9⇒(x2+x21−2×x×x1)−9⇒(x−x1)2−(3)2⇒(x−x1+3)(x−x1−3).
Hence, x2+1x2−11=(x−1x+3)(x−1x−3)x^2 + \dfrac{1}{x^2} - 11 = \Big(x - \dfrac{1}{x} + 3\Big) \Big(x - \dfrac{1}{x} - 3\Big)x2+x21−11=(x−x1+3)(x−x1−3).
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