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  4. If the LCM of the polynomials x5+x4+x2+1 and x4-1 is x8-1, then their mathrmHCF is

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Q. If the LCM of the polynomials $x^5+x^4+x^2+1$ and $x^4-1$ is $x^8-1$, then their $\mathrm{HCF}$ is

Polynomials

Solution:

We have, $P(x) Q(x)=\mathrm{LCM} \times \mathrm{HCF}$
$\left(x^8+x^4+x^2+1\right)\left(x^4-1\right)=\left(x^8-1\right) \times \mathrm{HCF} $
$\therefore \mathrm{HCF} =\frac{\left(x^8+x^4+x^2+1\right)-\left(x^4-1\right)}{\left(x^4-1\right)\left(x^4+1\right)}$
$ =\frac{x^8+x^4+x^2+1}{x^4+1}$