How do you express $\frac{x^{3} + 1}{x^{2} + 3}$ $(x^3 +1)/ (x^2 +3)$ in partial fractions?

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How do you express $\frac{x^{3} + 1}{x^{2} + 3}$ $(x^3 +1)/ (x^2 +3)$ in partial fractions?

1 Answer

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Answer 1 · 2016-09-17T08:16:54

$\frac{x^{3} + 1}{x^{2} + 3} = x - \frac{3 x - 1}{x^{2} + 3}$ $(x^3+1)/(x^2+3) = x - (3x-1)/(x^2+3)$

Explanation:

Note that $x^{2} + 3 \geq 3 > 0$ $x^2+3 >= 3 > 0$ for any Real values of $x$ $x$ , so the denominator has no Real zeros and we cannot break it down into linear factors (assuming that we want to stay with Real coefficients).

So the form of solution we are looking for is: $p (x) + \frac{A x + B}{x^{2} + 3}$ $p(x) + (Ax+B)/(x^2+3)$ for some polynomial $p (x)$ $p(x)$ and constants $A, B$ $A, B$ ...

$\frac{x^{3} + 1}{x^{2} + 3} = \frac{(x^{3} + 3 x) - (3 x - 1)}{x^{2} + 3} = x - \frac{3 x - 1}{x^{2} + 3}$ $(x^3+1)/(x^2+3) = ((x^3+3x)-(3x-1))/(x^2+3) = x - (3x-1)/(x^2+3)$

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How do you express $\frac{x^{3} + 1}{x^{2} + 3}$ $(x^3 +1)/ (x^2 +3)$ in partial fractions?

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