Find the GCD of the following $x^{3} - x^{2} + x - 1, x^{4} - 1$

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

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

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

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None of these

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Solution

The correct option is B $(x - 1) (x^{2} + 1)$
We first factorize the given polynomials $x^{3} - x^{2} + x - 1$ and $x^{4} - 1$ as shown below:

$x^{3} - x^{2} + x - 1 = x^{2} (x - 1) + 1 (x - 1) = (x - 1) (x^{2} + 1)$

$x^{4} - 1 = (x^{2})^{2} - (1)^{2} = (x^{2} - 1) (x^{2} + 1) (∵ a^{2} - b^{2} = (a + b) (a - b)) = (x - 1) (x + 1) (x^{2} + 1)$

The common factors of $x^{3} - x^{2} + x - 1$ and $x^{4} - 1$ are $(x - 1)$ and $(x^{2} + 1)$ , therefore, the GCD is $(x - 1) (x^{2} + 1)$ .

Hence, the greatest common divisor is $(x - 1) (x^{2} + 1)$ .

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