Question

Find the remainder when x³+ 3x²+ 3x+ 1 is divided by
x+1
x-1
x+π
5+2x

Answer 1

Answer:

$\bf\text{Remainder theorem}$

$\boxed{\text{The remainer when P(x) is divided by (x-a) is P(a)}}$

$P(x)=x^3+3x^2+3x+1$

1.

$\text{The remainder when P(x) is divided by (x+1) is P(-1)}$

$P(-1)=(-1)^3+3(-1)^2+3(-1)+1$

$P(-1)=-1+3-3+1$

$P(-1)=0$

2.

$\text{The remainder when P(x) is divided by (x-1) is P(1)}$

$P(1)=(1)^3+3(1)^2+3(1)+1$

$P(1)=1+3+3+1$

$P(1)=8$

3.

$\text{The remainder when P(x) is divided by }(x+\pi)\:is\:P(-\pi)$

$P(-\pi)=(-\pi)^3+3(\pi)^2+3(\pi)+1$

$P(-\pi)=-{\pi}^3+3(\pi)^2+3(\pi)+1$

4.

$\text{The remainder when P(x) is divided by (2x+5) is }P(\frac{-5}{2})$

$P(\frac{-5}{2})=(\frac{-5}{2})^3+3(\frac{-5}{2})^2+3(\frac{-5}{2})+1$

$P(\frac{-5}{2})=\frac{-125}{8}+3(\frac{25}{4})-\frac{15}{2}+1$

$P(\frac{-5}{2})=\frac{-125+150-60+8}{8}$

$P(\frac{-5}{2})=\frac{-27}{8}$