Question

Find the remainder when $x^3+3x^2+3x+1$ is divided by $x-\frac{1}{2}$.

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 $x-\frac{1}{2}$.

Put divisor $x-\frac{1}{2}$$=0$

$\Rightarrow$ $x=\frac{1}{2}$

Step 2: Determine the Remainder

Consider the equation

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

Putting $x=\frac{1}{2}$

$\therefore$Remainder is

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

$=\frac{1}{8}+\frac{3}{4}+\frac{3}{2}+1$

$=\frac{1+6+12+8}{8}$

$=\frac{27}{8}$

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