Question

Question 1
Find the remainder when $x^3+3x^2+3x+1$ is divided by
(i) x + 1
(ii) $x-\frac{1}{2}$
(iii) x
(iv) $x+\pi$
(v) 5 + 2x

Answer 1

$p\left(x\right)=x^3+3x^2+3x+1$
(i) When p(x) is divided by x + 1, the remainder is p(-1),
$p(-1)=(-1{)}^3+3(-1{)}^2+3(-1)+1=-1+3-3+1=0$
Remainder = 0

(ii) When p(x) is divided by $x-\frac{1}{2}$, the remainder is $p\left(\frac{1}{2}\right).p\left(\frac{1}{2}\right)={\left(\frac{1}{2}\right)}^3+3\left(\frac{1}{4}\right)+3\left(\frac{1}{2}\right)+1$
$=\frac{1}{8}+\frac{3}{4}+\frac{3}{2}+1=\frac{1+6+12+8}{8}$
$\therefore$ Remainder $=\frac{27}{8}=3\frac{3}{8}$

(iii) When p(x) is divided by x, then remainder is p(0).
x = 0, substitute in p(x).
$p\left(0\right)=0^3+3\times 0^2+3\times 0+1=1.$
$\therefore$ Remainder = 1

(iv) When p(x) is divided by $x+\pi ,$, then the remainder is $p(-\pi ).x=-\pi$ to be substituted in $p\left(x\right).p(-\pi )=(-\pi {)}^3+3(-\pi {)}^2+3(-\pi )+1.$
$\therefore$ Remainder $=-{\pi }^3+3{\pi }^2-3\pi +1$

(v) When p(x) is divided by (5 + 2x), then remainder is
$p\left(\frac{-5}{2}\right).p\left(\frac{-5}{2}\right)={\left(\frac{-5}{2}\right)}^3+3{\left(\frac{-5}{2}\right)}^2+3\left(\frac{-5}{2}\right)+1=\frac{-125}{8}+\frac{75}{4}-\frac{15}{2}+1=\frac{-125+150-60+8}{8}Remainder=\frac{-35+8}{8}=\frac{-27}{8}$