Question

By Remainder Theorem find the remainder, when $\mathrm{p}\left(\mathrm{x}\right)$ is divided by $\mathrm{g}\left(\mathrm{x}\right)$, where $\mathrm{p}\left(\mathrm{x}\right)={\mathrm{x}}^3-2{\mathrm{x}}^2-4\mathrm{x}-1$, $\mathrm{g}\left(\mathrm{x}\right)=\mathrm{x}+1$

Answer 1

The remainder theorem says that if a polynomial $\mathrm{p}\left(\mathrm{x}\right)$ is divided by a linear factor $\mathrm{x}-\mathrm{a}$ then the remainder is obtained as $\mathrm{p}\left(\mathrm{a}\right)$ .

Given that , $\mathrm{p}\left(\mathrm{x}\right)={\mathrm{x}}^3-2{\mathrm{x}}^2-4\mathrm{x}-1$ and $\mathrm{g}\left(\mathrm{x}\right)=\mathrm{x}+1$

So , zero of $\mathrm{g}\left(\mathrm{x}\right)=-1$

On applying the remainder theorem

Remainder when $\mathrm{p}\left(\mathrm{x}\right)$ divided by $\mathrm{g}\left(\mathrm{x}\right)=p(-1)$

$p(-1)={\left(-1\right)}^3-2{\left(-1\right)}^2-4\left(-1\right)-1$

$=0$

Therefore, the remainder $=0$