Question

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

Answer 1

Step 1: Determine the value of $x$ for the divisor

Given equation is $x^3+3x^2+3x+1$ which is divided by $5+2x$.

Put divisor $=0$

$5+2x=0$

$\Rightarrow$ $2x=-5$

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

Step 2: Determine the Remainder

Consider the equation

$x^3+3x^2+3x+1$

Upon putting $x=-\frac{5}{2}$

$\therefore$Remainder is

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

$=\left(-\frac{1255}{8}\right)+{\left(\frac{75}{4}\right)}^2-\left(\frac{15}{2}\right)+1$

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

Hence, remainder when $x^3+3x^2+3x+1$ is divided by $5+2x$ is $-\frac{27}{8}$.