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# What is 4-x^2+x^4 divided by x^2+x+5?

Jul 27, 2018

$\frac{4 - {x}^{2} + {x}^{4}}{{x}^{2} + x + 5} = {x}^{2} - x - 5 + \frac{10 x + 29}{{x}^{2} + x + 5}$

#### Explanation:

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

$= \frac{{x}^{4} - {x}^{2} + 4}{{x}^{2} + x + 5}$

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

$= \frac{{x}^{2} \left({x}^{2} + x + 5\right) - {x}^{3} - 6 {x}^{2} + 4}{{x}^{2} + x + 5}$

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

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

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

$= {x}^{2} - x + \frac{- 5 {x}^{2} + 5 x + 4}{{x}^{2} + x + 5}$

$= {x}^{2} - x + \frac{- 5 {x}^{2} - 5 x - 25 + 10 x + 29}{{x}^{2} + x + 5}$

$= {x}^{2} - x + \frac{- 5 \left({x}^{2} + x + 5\right) + 10 x + 29}{{x}^{2} + x + 5}$

$= {x}^{2} - x - 5 + \frac{10 x + 29}{{x}^{2} + x + 5}$

Alternatively, we can find the same result by long dividing tuples representing the coefficients of the powers of $x$.

In our example, we want to divide "$1 \setminus 0 \setminus - 1 \setminus 0 \setminus 4$" by "$1 \setminus 1 \setminus 5$". Note well the inclusion of $0$'s in the dividend, to represent the missing powers of $x$.

So ${x}^{4} - {x}^{2} + 4$ divided by ${x}^{2} + x + 5$ is ${x}^{2} - x - 5$ with remainder $10 x + 29$.