How do you express $\frac{x^{2} + 33}{x^{3} + x^{2}}$ $(x^2 + 33)/(x^3 + x^2)$ in partial fractions?

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Answer 1 · 2016-03-08T18:13:01

$\frac{x^{2} + 33}{x^{2} (x + 1)} = - \frac{33}{x} + \frac{33}{x^{2}} + \frac{34}{x + 1}$ $(x^2+33)/(x^2(x+1)) = -33/x +33/x^2 +34/(x+1)$

$\frac{x^{2} + 33}{x^{2} (x + 1)} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x + 1}$ $(x^2+33)/(x^2(x+1)) = A/x +B/x^2 +C/(x+1)$

$x^{2} + 33 = A (x (x + 1)) + B (x + 1) + C x^{2}$ $x^2+33 = A(x(x+1))+B(x+1)+Cx^2$

$x^{2} + 33 = A x^{2} + A x + B x + B + C x^{2}$ $x^2+33 = Ax^2+Ax+Bx+B+Cx^2$

$1 = A + C, 0 = A + B, 33 = B$ $1=A+C, 0=A+B, 33=B$

$A = - 33, B = 33, C = 34$ $A=-33, B=33, C=34$

$\frac{x^{2} + 33}{x^{2} (x + 1)} = - \frac{33}{x} + \frac{33}{x^{2}} + \frac{34}{x + 1}$ $(x^2+33)/(x^2(x+1)) = -33/x +33/x^2 +34/(x+1)$

How do you express (x^2 + 33)/(x^3 + x^2)x2+33x3+x2(x^2 + 33)/(x^3 + x^2) in partial fractions?