# How do you find the remainder when (2x^4+6x^3+5x-6) / (x+2)?

We can use synthetic division to find remainder or
We can also use remainder theorem
Remainder $= - 32$

#### Explanation:

$\frac{2 {x}^{4} + 6 {x}^{3} + 5 x - 6}{x + 2}$
Use synthetic Division

${x}^{4} \text{ " " " " "x^3" " " " " "x^2" " " " " "x^1" " " " } {x}^{0}$
$2 \text{ " " " " "6" " " " " "0" " " " " "5" " "" " "-6" " }$ trial divisor$= - 2$
$\underline{\text{ " " " " "-4 " " "" " "-4" " " " " "8" " " " } - 26}$
$2 \text{ " " " " " 2 " " " " ""-4 " " " " " "13 " " "" " } - 32$

Remainder$= - 32$

Also , we can use the remainder theorem
We will use $x = - 2$ in the dividend

Let $P \left(x\right) = 2 {x}^{4} + 6 {x}^{3} + 5 x - 6$
Find $P \left(- 2\right)$

$P \left(- 2\right) = 2 \cdot {\left(- 2\right)}^{4} + 6 {\left(- 2\right)}^{3} + 5 \left(- 2\right) - 6$

$P \left(- 2\right) = - 32 \text{ " }$and this is the remainder