Question

$x^6 - 1 \div x {2} - 1$
solv que.

Answer 1

solution:

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Given:

$\frac{x^6 -1}{x^2-1}$

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we know that,

a² - b² = (a+b) (a-b)

and 1 power anything is 1,

so,

$x^6 - 1 = (x^3 + 1)(x^3 - 1)$

x² - 1 = (x+1)(x-1)

By using

a³ + b³ = (a+b) (a² - ab +b²)
a³ - b³ = (a-b) (a² + ab +b²)

(x³+1) =(x+1)(x²- x + 1)

& (x³-1) = (x-1) (x²+x+1)

so,

=> $\frac{x^6-1}{x^2-1} = \frac{(x^3 +1)(x^3-1)}{(x+1)(x-1)}$


=> $\frac{x^6 -1}{x^2-1} = \frac{(x+1)(x^2-x+1)(x-1)(x^2 +x +1)}{(x-1)(x+1)}$

=> $\frac{x^6 - 1}{x^2-1} =(x^2 -x+1)(x^2+x+1)$

=>$\frac{x^6 - 1}{x^2 - 1} = [(x^2 + 1) -x] [(x^2+1)+x]$

=> $\frac{x^6-1}{x^2-1} = (x^2+1)^2-(x)^2$

=> $\frac{x^6 - 1}{x^2-1} = x^4 +2x^2 +1 -x^2$

=>$\frac{x^6 -1}{x^2 - 1} = x^4 + x^2 + 1$

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                                                    Hope it Helps!!