# How do you use synthetic division to divide y^2 + 25 by y+5?

Sep 2, 2015

Write ${y}^{2} + 25$ as $1 , 0 , 25$ (not forgetting the $0$) and $y + 5$ as $1 , 5$, then proceed in a similar fashion to long division to find quotient $1 , - 5$ meaning $y - 5$ and remainder $50$

#### Explanation:

${y}^{2} + 25 = {y}^{2} + 0 y + 25$ is represented by the sequence $1$, $0$, $25$.

$y + 5$ is represented by the sequence $1$, $5$.

Write out like long division of integers and proceed similarly:

Write $1$, $0$, $25$ under the bar as the dividend and $1$, $5$ to the left as the divisor.

Identify $1$ as the first term of the quotient - writing it above the bar - choosing it to cause the leading terms to match when the divisor is multiplied by it.

Write out $1 \times \left(1 , 5\right)$ under the dividend and subtract it to get the first term $- 5$ of a remainder.

Bring down the next term $25$ of the dividend alongside it.

Identify $- 5$ as the second term of the quotient - writing above the bar - choosing it to cause the leading terms to match when the divisor is multiplied by it.

Write out $- 5 \times \left(1 , 5\right)$ under the remainder and subtract it to get the remainder $50$.

We stop with this remainder since there are not enough terms remaining to make a sequence with enough terms to be divisible by the divisor.

So we have found:

$\frac{{y}^{2} + 25}{y + 5} = \left(y - 5\right) + \frac{50}{y + 5}$

or if you prefer:

${y}^{2} + 25 = \left(y + 5\right) \left(y - 5\right) + 50$