# How do you divide ( x^5 - x^3 - x^2 - 17x - 15 )/(x^2 - 2 )?

Jan 5, 2016

Long divide the coefficients to find:

$\frac{{x}^{5} - {x}^{3} - {x}^{2} - 17 x - 15}{{x}^{2} - 2} = \left({x}^{3} + x - 1\right) + \frac{- 15 x + 13}{{x}^{2} - 2}$

...not forgetting to include $0$'s for any missing powers of $x$.

#### Explanation:

I like to long divide the coefficients, not forgetting to include $0$'s for any missing powers of $x$...

This is similar to long division of numbers.

Reconstructing polynomials from the resulting sequences we find that the quotient is ${x}^{3} + x - 1$ with remainder $- 15 x + 13$

That is:

$\frac{{x}^{5} - {x}^{3} - {x}^{2} - 17 x - 15}{{x}^{2} - 2} = \left({x}^{3} + x - 1\right) + \frac{- 15 x + 13}{{x}^{2} - 2}$

Or if you prefer:

$\left({x}^{5} - {x}^{3} - {x}^{2} - 17 x - 15\right) = \left({x}^{2} - 2\right) \left({x}^{3} + x - 1\right) + \left(- 15 x + 13\right)$