# How do you use synthetic division to divide x^3+5x^2-x-9 by x+2?

Oct 6, 2015

See explanation...

#### Explanation:

Synthetic division of polynomials is similar to long division of numbers... Write the coefficients of the dividend $1 , 5 , - 1 , 9$ under the bar.

Write the coefficients of the divisor $1 , 2$ to the left of the bar.

Write the first coefficient $1$ of the quotient above the bar, in order that when the divisor is multiplied by this number, the result will match the dividend in its first term.

Write the result $1 , 2$ of multiplying the divisor by the first term $1$ of the quotient under the divisor.

Subtract this from the divisor to give the remainder $3$.

Write the next term $- 1$ from the divisor alongside it.

Write the next coefficient $3$ of the quotient above the bar, chosen so that when the divisor $1 , 2$ is multiplied by this term, the resulting leading term will match the first term of the remainder.

Continue in similar fashion, until there are no more terms of the dividend to bring down.

The final remainder is the remainder of the whole division.

In our case, we find that ${x}^{3} + 5 {x}^{2} - x - 9$ divided by $x + 2$ is ${x}^{2} + 3 x - 7$ with remainder $5$.

So: ${x}^{3} + 5 {x}^{2} - x - 9 = \left(x + 2\right) \left({x}^{2} + 3 x - 7\right) + 5$

or: $\frac{{x}^{3} + 5 {x}^{2} - x - 9}{x + 2} = {x}^{2} + 3 x - 7 + \frac{5}{x + 2}$