# How do you use the remainder theorem to find the remainder for the division (10x^3-11x^2-47x+30)div(x+2)?

Nov 7, 2016

The remainder states that when any polynomial $f \left(x\right)$ is divided by $x - a$, the remainder is $f \left(a\right)$.

So, Letting $f \left(x\right) = 10 {x}^{3} - 11 {x}^{2} - 47 x + 30$, we have:

$f \left(- 2\right) = 10 {\left(- 2\right)}^{3} - 11 {\left(- 2\right)}^{2} - 47 \left(- 2\right) + 30$

$f \left(- 2\right) = 10 \left(- 8\right) - 11 \left(4\right) + 94 + 30$

$f \left(- 2\right) = - 80 - 44 + 94 + 30$

$f \left(- 2\right) = 0$

The remainder is $0$. In other words, $x + 2$ is a factor of $10 {x}^{3} - 11 {x}^{2} - 47 x + 30$.

Hopefully this helps!

Nov 7, 2016

This expression has a remainder of zero.
$\frac{10 {x}^{3} - 11 {x}^{2} - 47 x + 30}{x + 2}$ is equal to $\left(10 {x}^{2} - 31 x + 15\right)$, with no remainder.