How do you express $\frac{x^{2}}{x^{2} + x + 2}$ $x^2/(x^2 + x +2)$ in partial fractions?

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How do you express $\frac{x^{2}}{x^{2} + x + 2}$ $x^2/(x^2 + x +2)$ in partial fractions?

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Answer 1 · 2016-02-15T08:14:28

Partial fractions of $\frac{x^{2}}{x^{2} + x + 2}$ $x^2/(x^2+x+2)$ are $1 - \frac{x + 2}{x^{2} + x + 2}$ $1-(x+2)/(x^2+x+2$

Explanation:

As the denominator $x^{2} + x + 2$ $x^2+x+2$ is quadratic and its determinant $\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ $(-b+-sqrt(b^2-4ac))/(2a)$ is not rational (as $\sqrt{b^{2} - 4 a c} = \sqrt{- 7}$ $sqrt(b^2-4ac)=sqrt(-7)$ is not rational), its partial fractions will be of type $\frac{A x + B}{x^{2} + x + 2}$ $(Ax+B)/(x^2+x+2)$ .

But, degree of numerator is $2$ $2$ hence let us write $\frac{x^{2}}{x^{2} + x + 2}$ $x^2/(x^2+x+2)$ as

$\frac{x^{2}}{x^{2} + x + 2} = 1 - \frac{x + 2}{x^{2} + x + 2}$ $x^2/(x^2+x+2)=1-(x+2)/(x^2+x+2$

Hence partial fractions of $\frac{x^{2}}{x^{2} + x + 2}$ $x^2/(x^2+x+2)$ are $1 - \frac{x + 2}{x^{2} + x + 2}$ $1-(x+2)/(x^2+x+2$

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How do you express $\frac{x^{2}}{x^{2} + x + 2}$ $x^2/(x^2 + x +2)$ in partial fractions?

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