Question

$p\left(x\right)=x^3+3x^2+3x+1$ is divided by $x=-\frac{1}{2}$
$p\left(a\right)x=-\frac{1}{2}=0$
$x=\frac{1}{2}$
$p\left(\frac{1}{2}\right)={\left(\frac{1}{2}\right)}^3+3{\left(\frac{1}{2}\right)}^2+3\left(\frac{1}{2}\right)+1$

Answer 1

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

$g\left(x\right)=x-\frac{1}{2}$

$g\left(x\right)=0,x=\frac{1}{2}$

by factor theorem when $p\left(x\right)$ is divided by $g\left(x\right)$

Remainder $=p\left(\frac{1}{2}\right)$

$(\frac{1}{2}{)}^3+3(\frac{1}{2}{)}^2+3\left(\frac{1}{2}\right)+1=\frac{27}{8}$