Question

Use Remainder theorem to factorize the following polynomial. $2x^3+3x^2-9x-10$.

Answer 1

Let $f\left(x\right)=2x^3+3x^2-9x-10$, then $f(-1)=-2+3+9-10=0$

So, by remainder theorem, when $f\left(x\right)$ is divided by $x+1$, the remainder is $0$. So, $(x+1)$ is a factor of $f\left(x\right)$

$f\left(2\right)=16+12-18-10=0$

So, by similar reasoning, $(x-2)$ is also a factor of $f\left(x\right)$

Therefore, $f\left(x\right)=(x+1)(x-2)p\left(x\right)=(x^2-x-2)p\left(x\right)$

By synthetic division, we can find $p\left(x\right)$

$\Rightarrow f\left(x\right)=2x^3+3x^2-9x-10=(x^2-x-2)(2x+5)$

Therefore $p\left(x\right)=2x+5$ and the factorization of $f\left(x\right)$ is

$f\left(x\right)=(x+1)(x-2)(2x+5)$