Question

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

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

Answer 1

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

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

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

$\Rightarrow \left(\right(x+1{)}^2(x+2)\left)\right(q\left(x\right))=(x+1{)}^3(x+2{)}^3$

$\Rightarrow q\left(x\right)=\frac{(x+1{)}^3(x+2{)}^3}{(x+1{)}^2(x+2)}$

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

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