Find the HCF of $f (x) = x^{3} + x^{2} - x - 1$ and $g (x) = x^{3} + x^{2} + x + 1$

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Solution

We know that $H C F$ is the highest common factor.

Factorise $x^{3} + x^{2} - x - 1$ as follows:

$x^{3} + x^{2} - x - 1 = x^{2} (x + 1) - 1 (x + 1) = (x^{2} - 1) (x + 1) = (x - 1) (x + 1) (x + 1) = (x - 1) (x + 1)^{2}$
(using identity $a^{2} - b^{2} = (a + b) (a - b)$ )

Now, factorise $x^{3} + x^{2} + x + 1$ as follows:

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

Since the common factor between the polynomials $x^{3} + x^{2} - x - 1$ and $x^{3} + x^{2} + x + 1$ is $(x + 1)$ .

Hence, the $H C F$ is $(x + 1)$ .

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