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Correct Marks 2

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Q 1. L.C.M. of {tex} x^{4}+x^{2}+1,\ x^{4}-x^{2}-2x-1,\ x^{6}-1 {/tex} is

A

{tex} \left(x^{6}+1\right) \left(x^{4}+x^{2}+1\right) {/tex}

B

{tex} \left(x^{6}-1\right) \left(x^{2}+x+1\right) {/tex}

C

{tex} \left(x^{6}+1\right) \left(x^{2}-x-1\right) {/tex}

{tex} \left(x^{6}-1\right) \left(x^{2}-x-1\right) {/tex}

Explanation

{tex} x^{4}+x^{2}+1 = x^{4}+2x^{2}+1-x^{2} {/tex}
= {tex} \left(x^{2}+1\right)^{2}-x^{2} {/tex}
= {tex} \left(x^{2}+1+x\right) \left(x^{2}+1-x\right) {/tex} ...(i)
{tex} x^{4}-x^{2}-2x-1 = x^{4}- \left(x^{2}+2x+1\right) {/tex}
= {tex} \left(x^{2}\right)^{2}- \left(x+1\right)^{2} {/tex}
= {tex} \left(x^{2}+x+1\right) \left(x^{2}-x-1\right) {/tex} ...(ii)
{tex} x^{6}-1 = \left(x^{3}-1\right) \left(x^{3}+1\right) {/tex}
= {tex} \left(x-1\right) \left(x^{2}+x+1\right) \left(x+1\right) \left(x^{2}-x+1\right) {/tex} ...(iii)
L.C.M. = {tex} \left(x^{3}-1\right) \left(x^{3}+1\right) \left(x^{2}-x-1\right) {/tex}
= {tex} \left(x^{6}-1\right) \left(x^{2}-x-1\right) {/tex}