How do you use the remainder theorem to determine the remainder when the polynomial x^4+x^3-5x^2+2x-7 is divided by x+2?

Dec 23, 2016

The remainder is the result of substituting $x = - 2$, namely $- 23$

Explanation:

$f \left(x\right) = {x}^{4} + {x}^{3} - 5 {x}^{2} + 2 x - 7$

The remainder when divided by $\left(x - a\right)$ is $f \left(a\right)$.

So the remainder when divided by $\left(x + 2\right)$ is:

$f \left(- 2\right) = 16 - 8 - 20 - 4 - 7 = - 23$

Here is a long division of the coefficients, just to check:

So:

${x}^{4} + {x}^{3} - 5 {x}^{2} + 2 x - 7 = \left({x}^{3} - {x}^{2} - 3 x + 8\right) \left(x + 2\right) - 23$