Question

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

Answer 1

`x^2/(x+1)^3=1/(x+1)^3-2/(x+1)^2+1/(x+1)`

Explanation:

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

`x^2/(x+1)^3=A/(x+1)^3+B/(x+1)^2+C/(x+1)`

`x^2/(x+1)^3=A/(x+1)^3+(B(x+1))/(x+1)^3+(C(x+1)^2)/(x+1)^3`

`x^2/(x+1)^3=(A+B(x+1)+C(x+1)^2)/(x+1)^3`

`x^2/(x+1)^3=(A+Bx+B+Cx^2+2Cx+C)/(x+1)^3`

`(x^2+0*x+0*x^0)/(x+1)^3=(Cx^2+Bx+2Cx+(A+B+C)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

`Cx^2=1*x^2`

`Bx+2Cx=0*x`

`(A+B+C)x^0=0*x^0`

Therefore, after simplification, the equations are:

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

Solving simultaneously, the values are

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

so that

`x^2/(x+1)^3=1/(x+1)^3-2/(x+1)^2+1/(x+1)`

have a nice day ! from the Philippines..