Question

Find the remainder when $x^3+3x^2+3x+1$ is divided by
$5+2x$

Answer 1

Dividing $x^3+3x^2+3x+1$ by $5+2x$

So, $x^3+3x^2+3x+1$ is dividend and $5+2x$ is divisor.

Put, $5+2x=0$

$\therefore$ $x=\frac{-5}{2}$

Let $p\left(x\right)=x^3+3x^2+3x+1$

Putting $x=\frac{-5}{2}$

$p\left(\frac{-5}{2}\right)={\left(\frac{-5}{2}\right)}^3+3{\left(\frac{-5}{2}\right)}^2+3\left(\frac{-5}{2}\right)+1$

$=\frac{-125}{8}+\frac{75}{4}-\frac{15}{2}+1$

$=\frac{-125+150-60+8}{8}$

$=\frac{-27}{8}$

Thus, reminder of $p\left(\frac{-5}{2}\right)=\frac{-27}{8}$.

$\therefore$ We will get reminder as $\frac{-27}{8}$ when $x^3+3x^2+3x+1$ is divided by $5+2x$.