# How do you use the remainder theorem to find the remainder for each division (x^5+32)div(x+2)?

Dec 17, 2016

## 0

#### Explanation:

the Remainder theorem states :

if a polynomial $\text{ "P(x)" }$is divided by $\text{ "(x-a)" }$ the remainder is $\text{ "P(a)" }$

proof:

$P \left(x\right) = \left(x - a\right) Q \left(x\right) + R$

$P \left(a\right) = \cancel{\left(a - a\right) Q \left(x\right)} + R$

$\therefore P \left(a\right) = R$

$\left({x}^{5} + 32\right) \div \left(x + 2\right)$

$P \left(x\right) = \left({x}^{5} + 32\right)$

to find remainder

$P \left(- 2\right) = {\left(- 2\right)}^{5} + 32 = - 32 + 32 = 0$

remainder $\text{ "=0" }$which implies $\text{ "(x+2)" }$is a factor of$\text{ } \left({x}^{5} + 32\right)$