# How do you divide (x^4-5x^3+2x^2-4) / (x^3-4x)  using polynomial long division?

Sep 5, 2016

$x - 5 + \frac{6 {x}^{2} - 20 x - 4}{{x}^{3} - 4 x}$

#### Explanation:

Incorporating place keepers such as $0 x$ so that things line up properly.

$\text{ } {x}^{4} - 5 {x}^{3} + 2 {x}^{2} + 0 x - 4$
$\textcolor{m a \ge n t a}{x} \left({x}^{3} - 4 x\right) \to \text{ "ul( x^4+0x^3-4x^2) larr" Subtract}$
$\text{ } 0 - 5 {x}^{3} + 6 {x}^{2} + 0 x - 4$
$\textcolor{m a \ge n t a}{- 5} \left({x}^{3} - 4 x\right) \to \text{ "color(white)()ul(-5x^3+0x^2+20x ) larr" Subtract}$
$\text{ "color(blue)(0 +6x^2-20x-4) larr" Remainder}$

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Highest order of $6 {x}^{2} - 20 x - 4$ is 2 from $6 {x}^{2}$

Highest order of the divisor ${x}^{3} - 4 x$

As the order of the divisor is now the highest we stop and we have to express the remainder in fraction form with the divisor as the denominator giving:

(x^4-5x^3+2x^2-4) -: (x^3-4x) = color(magenta)(x-5 +(color(blue)(6x^2-20x-4))/(x^3-4x)