Factorise the following:
x3+x2−1x2+1x3x^3 + x^2 - \dfrac{1}{x^2} + \dfrac{1}{x^3}x3+x2−x21+x31
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x3+x2−1x2+1x3=x3+1x3+x2−1x2x^3 + x^2 - \dfrac{1}{x^2} + \dfrac{1}{x^3} = x^3 + \dfrac{1}{x^3} + x^2 - \dfrac{1}{x^2}x3+x2−x21+x31=x3+x31+x2−x21
We know that,
a2 - b2 = (a + b)(a - b)
a3 + b3 = (a + b)(a2 - ab + b2)
∴x3+1x3+x2−1x2=(x+1x)(x2−x×1x+1x2)+(x+1x)(x−1x)=(x+1x)(x2−1+1x2+x−1x).\therefore x^3 + \dfrac{1}{x^3} + x^2 - \dfrac{1}{x^2} = \Big(x + \dfrac{1}{x}\Big)\Big(x^2 - x \times \dfrac{1}{x} + \dfrac{1}{x^2}\Big) + \Big(x + \dfrac{1}{x}\Big)\Big(x - \dfrac{1}{x}\Big) \\[1em] = \Big(x + \dfrac{1}{x}\Big)\Big(x^2 - 1 + \dfrac{1}{x^2} + x - \dfrac{1}{x}\Big).∴x3+x31+x2−x21=(x+x1)(x2−x×x1+x21)+(x+x1)(x−x1)=(x+x1)(x2−1+x21+x−x1).
Hence, x3+x2−1x2+1x3=(x+1x)(x2−1+1x2+x−1x).x^3 + x^2 - \dfrac{1}{x^2} + \dfrac{1}{x^3} = \Big(x + \dfrac{1}{x}\Big)\Big(x^2 - 1 + \dfrac{1}{x^2} + x - \dfrac{1}{x}\Big).x3+x2−x21+x31=(x+x1)(x2−1+x21+x−x1).
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