# How do you express (x^2)/(x+1)^3 in partial fractions?

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

#### Explanation:

Set up the equation first by assigning letter variables A, B, C.

${x}^{2} / {\left(x + 1\right)}^{3} = \frac{A}{x + 1} ^ 3 + \frac{B}{x + 1} ^ 2 + \frac{C}{x + 1}$

${x}^{2} / {\left(x + 1\right)}^{3} = \frac{A}{x + 1} ^ 3 + \frac{B \left(x + 1\right)}{x + 1} ^ 3 + \frac{C {\left(x + 1\right)}^{2}}{x + 1} ^ 3$

${x}^{2} / {\left(x + 1\right)}^{3} = \frac{A + B \left(x + 1\right) + C {\left(x + 1\right)}^{2}}{x + 1} ^ 3$

${x}^{2} / {\left(x + 1\right)}^{3} = \frac{A + B x + B + C {x}^{2} + 2 C x + C}{x + 1} ^ 3$

$\frac{{x}^{2} + 0 \cdot x + 0 \cdot {x}^{0}}{x + 1} ^ 3 = \frac{C {x}^{2} + B x + 2 C x + \left(A + B + C\right) {x}^{0}}{x + 1} ^ 3$

Set up the equations to solve for the values of A, B, C by equating the coefficients of each term

$C {x}^{2} = 1 \cdot {x}^{2}$

$B x + 2 C x = 0 \cdot x$

$\left(A + B + C\right) {x}^{0} = 0 \cdot {x}^{0}$

Therefore, after simplification, the equations are:

$C = 1$
$B + 2 C = 0$
$A = B + C = 0$

Solving simultaneously, the values are

$A = 1$ and $B = - 2$ and $C = 1$

so that

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

have a nice day ! from the Philippines..