# How do you divide (2x^2-2x+4)/( 2x-1)?

Nov 5, 2015

$x - \left(\frac{3}{2}\right) + \frac{5}{4 x - 2}$

#### Explanation:

Polynomial division works much the same way as regular long division:

Find a value that multiplied to the first term of the denominator is equal to the first term of the numerator (sorry, formatting this is a little hard)
$\left(2 x - 1\right) \cdot x = 2 {x}^{2} - x$

so your first term is x

next subtract the numerator by new value
$\left(2 {x}^{2} - 2 x + 4\right) - \left(2 {x}^{2} - x\right) = - 3 x + 4$

Then repeat the first term for this new value, find a value that multiplied by your (original) denominator is equal to the new value
$\left(2 x - 1\right) \cdot - \frac{3}{2} = - 3 x + \left(\frac{3}{2}\right)$

so your second term is $- \left(\frac{3}{2}\right)$

subtract the numerator (from after the first subtraction) by this new value
$\left(- 3 x + 4\right) - \left(- 3 x + \left(\frac{3}{2}\right)\right) = \frac{5}{2}$

Finally like the remainder of normal long division you make a fraction over the divisor
$\frac{\frac{5}{2}}{2 x - 1} = \frac{5}{4 x - 2}$

This is your last term leaving you with
$x - \left(\frac{3}{2}\right) + \frac{5}{4 x - 2}$

A great resource for these sorts of problems if you can't understand my formatting: http://www.purplemath.com/modules/polydiv2.htm