# How do you divide (8x^3+5x^2-12x+10)/(x^2-3)?

May 13, 2016

$\frac{8 {x}^{3} + 5 {x}^{2} - 12 x + 10}{{x}^{2} - 3} = 8 x + 5 + \frac{12 x + 25}{{x}^{2} - 3}$

#### Explanation:

I like to long divide just the coefficients, not forgetting to include $0$'s for any missing powers of $x$. In our current example that means the missing term in $x$ in the divisor... The process is similar to long division of numbers.

Write the dividend under the bar and the divisor to the left.

Write the first term $\textcolor{b l u e}{8}$ of the quotient above the bar, choosing it so that when multiplied by the divisor $1 , 0 , - 3$ the product matches the first term of the dividend.

Write the product $8 , 0 , - 24$ of $8$ and the divisor under the dividend and subtract it to give a remainder $5 , 12$.

Bring down the next term from the dividend alongside it, then choose the next term $\textcolor{b l u e}{5}$ of the quotient to match the leading term of our running remainder.

Write the product $5 , 0 , - 15$ under the running remainder and subtract it to give the final remainder $12 , 25$.

There are no more terms to bring down from the dividend and the remainder is now shorter than the divisor, so this is where we stop.

Our quotient is $8 , 5$, meaning $8 x + 5$ and our final remainder is $12 , 25$ meaning $12 x + 25$

So:

$\frac{8 {x}^{3} + 5 {x}^{2} - 12 x + 10}{{x}^{2} - 3} = 8 x + 5 + \frac{12 x + 25}{{x}^{2} - 3}$