Question

Find the LCM of the following: $2x^3+15x^2+2x-35,x^3+8x^2+4x-21$ whose GCD is $x+7$.

• A. $(2x^2+x-5)(x^3+8x^2+4x-21)$
• B. $(2x^2-x-5)(x^3+8x^2+4x-21)$
• C. $(2x^2+x-5)(x^3-8x^2+4x-21)$
• D. None of these

Answer 1

The correct option is A $(2x^2+x-5)(x^3+8x^2+4x-21)$

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 the GCD of the polynomials $2x^3+15x^2+2x-35$ and $x^3+8x^2+4x-21$ is $(x+7)$, therefore, we have:

$(2x^3+15x^2+2x-35)(x^3+8x^2+4x-21)=(x+7)\times LCM$

$\Rightarrow LCM=\frac{(2x^3+15x^2+2x-35)(x^3+8x^2+4x-21)}{(x+7)}$

To find the LCM, we have to do the long division as shown in the above image:

On dividing $2x^3+15x^2+2x-35$ by $(x+7)$, the quotient is $2x^2+x-5$ and the remainder is $0$, therefore,

$LCM=(2x^2+x-5)(x^3+8x^2+4x-21)$

Hence, the LCM of $2x^3+15x^2+2x-35$ and $x^3+8x^2+4x-21$ is $(2x^2+x-5)(x^3+8x^2+4x-21)$.