UPSC CDS 2 2022 Math Paper

We have,
p(x)=x4+x2+1
Using the formula:
a4+a2+1=(a2−a+1)(a2+a+1)
(a+b+c)2=a2+b2+c2+2ab+2bc+2ac
⇒p(x)=(x2−x+1)(x2+x+1)‌‌‌⋅⋅⋅⋅⋅⋅⋅(i)
⇒q(x)=x4−2x3+3x2−2x+1‌‌‌⋅⋅⋅⋅⋅⋅⋅(ii)
Also, we have, (x2−x+1)
By using the above formula
⇒(x2−x+1)2=x4+x2+1+2(−x)3+(−2x)+(2x2)
⇒(x2−x+1)2=x4+3x2−2x3−2x+1
Using equation (2)
⇒q(x)=(x2−x+1)2
Now we have two polynomial p(x)&q(x)
By using the concept of LCM discussed above
LCM=(x2+x+1)(x2−x+1)2
∴ The required LCM is (x2+x+1)(x2−x+1)2