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

Question

What does partial-fraction decomposition mean?
What is the partial-fraction decomposition of $\frac{5 x + 7}{x^{2} + 4 x - 5}$ $(5x+7)/(x^2+4x-5)$ ?
What is the partial-fraction decomposition of $\frac{x + 11}{(x + 3) (x - 5)}$ $(x+11)/((x+3)(x-5))$ ?
What is the partial-fraction decomposition of $\frac{x^{2} + 2 x + 7}{x {(x - 1)}^{2}}$ $(x^2+2x+7)/(x(x-1)^2)$ ?
How do you write $\frac{2}{x^{3} - x^{2}}$ $2/(x^3-x^2)$ as a partial fraction decomposition?
How do you write $\frac{x^{4}}{{(x - 1)}^{3}}$ $x^4/(x-1)^3$ as a partial fraction decomposition?
How do you write $\frac{3 x}{(x + 2) (x - 1)}$ $(3x)/((x + 2)(x - 1))$ as a partial fraction decomposition?
How do you write the partial fraction decomposition of the rational expression $\frac{x^{2}}{x^{2} + x + 4}$ $x^2/ (x^2+x+4)$ ?
How do you write the partial fraction decomposition of the rational expression # (3x^2 + 12x -...
How do you write the partial fraction decomposition of the rational expression $\frac{1}{(x + 6) (x^{2} + 3)}$ $1/((x+6)(x^2+3))$ ?

1423 views around the world

You can reuse this answer
Creative Commons License

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 x2+33x3+x2(x^2 + 33)/(x^3 + x^2) in partial fractions?