Question

Find the other polynomial q (x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively. $2(x+1)(x^2-4),(x+1),(x+1)(x-2)$

• A. $2(x+1)(x+2)$
• B. $2(x-1)(x+2)$
• C. $2(x+1)(x-2)$
• D. None of these

Answer 1

The correct option is A $2(x+1)(x+2)$

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 $p\left(x\right)=(x+1)(x-2)$, the LCM is $2(x+1)(x^2-4)$ and the GCD is $x+1$, therefore, we have:

$\left(\right(x+1\left)\right(x-2\left)\right)\left(q\right(x\left)\right)=(x+1)\times 2(x+1)(x^2-4)$

$\Rightarrow \left(\right(x+1\left)\right(x-2\left)\right)\left(q\right(x\left)\right)=2(x+1{)}^2\times (x^2-2^2)$

$\Rightarrow \left(\right(x+1\left)\right(x-2\left)\right)\left(q\right(x\left)\right)=2(x+1{)}^2\times (x+2\left)\right(x-2\left)\right(\because (a+b{)}^2=a^2+b^2+2ab)$

$\Rightarrow \left(\right(x+1\left)\right(x-2\left)\right)\left(q\right(x\left)\right)=2(x+1{)}^2(x+2\left)\right(x-2)$

$\Rightarrow q\left(x\right)=\frac{2(x+1{)}^2(x+2\left)\right(x-2)}{(x+1)(x-2)}$

$\Rightarrow q\left(x\right)=2(x+1)(x+2)$

Hence, the other polynomial $q\left(x\right)$ is $2(x+1)(x+2)$.