# How do you express x^2 /(x^2 +x+2) in partial fractions?

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Nimo N. Share
Mar 8, 2018

See below.

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Question:
How do you express
 color(blue)( (x^2)/(x^2 + x + 2)  in partial fractions?

Given: $\frac{{x}^{2}}{{x}^{2} + x + 2}$. Note that the degrees of the numerator and denominator are the same. And, as a bonus, the denominator is not factorable with real numbers.

The discriminant for the denominator:
 color(red)( b^2 - 4ac = (1)^2 - 4(1)(2) = 1 - 8 = -7  $, \setminus$ tells the story.

As in reference A), above, a case such as ours, the procedure is to do long division to obtain a fraction that might be written in partial fraction form.

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus 1$
${x}^{2} + x + 2 \overline{\text{)} {x}^{2}}$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus {x}^{2} + x + 2$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \text{---------------}$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus - x - 2$

So,
 color(red)( (x^2)/(x^2 + x + 2) = 1 - (x + 2)/(x^2 + x + 2)

Concentrate on the fraction, then we can put it back together. Since the denominator can not be factored with real numbers, we leave it alone and write:
$\frac{x + 2}{{x}^{2} + x + 2} = \frac{A x + B}{{x}^{2} + x + 2}$.

Then, solve for A and B.
The denominators are the same, so the numerators must be equal.
$\left(x + 2\right) = \left(A x + B\right)$
The only way that can happen is if A = 1, and B = 2.

Putting it back together:
 color(brown)( (x^2)/(x^2 + x + 2) = 1 - (1x + 2)/(x^2 + x + 2) .
This expression looks familiar, but it is all we can do!

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