Question

Find the other polynomial q (x) of the following, given that LCM, GCD and one polynomial p(x) respectively. $(x^4-y^4)(x^4+x^2{y}^2+y^4),x^2-y^2,x^4-y^4$

• A. $(x^2-y^2)(x^4+x^2{y}^2+y^4)$
• B. $(x^2+y^2)(x^4+x^2{y}^2+y^4)$
• C. $(x^2-y^2)(x^4-x^2{y}^2+y^4)$
• D. None of these

Answer 1

The correct option is A $(x^2-y^2)(x^4+x^2{y}^2+y^4)$

We know that if $p\left(x\right)$ and $q\left(x\right)$ are two polynomials, then $p\left(x\right)\times q\left(x\right)=$ {GCD of $p\left(x\right)$ and $q\left(x\right)$}$\times$ {LCM of $p\left(x\right)$ and $q\left(x\right)$}.

Now, it is given that one of the polynomial is $p\left(x\right)=x^4-y^4$, the LCM is $(x^4-y^4)(x^4+x^2{y}^2+y^4)$ and the GCD is $x^2-y^2$, therefore, we have:

$(x^4-y^4)\left(q\right(x\left)\right)=(x^2-y^2)\times (x^4-y^4)(x^4+x^2{y}^2+y^4)$

$\Rightarrow (x^4-y^4)\left(q\right(x\left)\right)=(x^2-y^2)(x^4-y^4)(x^4+x^2{y}^2+y^4)$

$\Rightarrow q\left(x\right)=\frac{(x^2-y^2)(x^4-y^4)(x^4+x^2{y}^2+y^4)}{(x^4-y^4)}$

$\Rightarrow q\left(x\right)=(x^2-y^2)(x^4+x^2{y}^2+y^4)$

Hence, the other polynomial $q\left(x\right)$ is $(x^2-y^2)(x^4+x^2{y}^2+y^4)$.