# How do you use the remainder theorem and Synthetic Division to find the remainders in the following division problems -2x^4 - 6x^2 + 3x + 1  divided by x+1?

Jul 30, 2015

Let $f \left(x\right) = - 2 {x}^{4} - 6 {x}^{2} + 3 x + 1$.

Using the remainder theorem, the remainder is

$f \left(- 1\right) = - 2 - 6 - 3 + 1 = - 10$

Using Synthetic division we get the same remainder.

#### Explanation:

The remainder theorem states that the remainder of dividing a polynomial $f \left(x\right)$ by $\left(x - a\right)$ is $f \left(a\right)$. In our case $a = - 1$ and $f \left(- 1\right) = - 10$.

Alternatively, using synthetic division we get the same remainder...

Here we divide $- 2 {x}^{4} - 6 {x}^{2} + 3 x + 1$ by $x + 1$. Note the $0$ in $- 2 , 0 , - 6 , 3 , 1$, standing for the missing ${x}^{3}$ term's coefficient.