Question

How do you express `(x^2-16x+9)/(x^4+10x^2+9)` in partial fractions?

Answer 1

`(x^2-16x+9)/(x^4+10x^2+9) = (-2x+1)/(x^2+1) + (2x)/(x^2+9)`

Explanation:

`(x^2-16x+9)/(x^4+10x^2+9) = (x^2-16x+9)/((x^2+1)(x^2+9))`

`color(white)((x^2-16x+9)/(x^4+10x^2+9)) = (Ax+B)/(x^2+1) + (Cx+D)/(x^2+9)`

`color(white)((x^2-16x+9)/(x^4+10x^2+9)) = ((Ax+B)(x^2+9)+(Cx+D)(x^2+1))/(x^4+10x^2+9)`

`color(white)((x^2-16x+9)/(x^4+10x^2+9)) = ((A+C)x^3+(B+D)x^2+(9A+C)x+(9B+D))/(x^4+10x^2+9)`

So equating coefficients, we get:

`{ (A+C = 0), (B+D = 1), (9A+C = -16), (9B+D = 9) :}`

Subtracting the first equation from the third, we get:

`8A = -16`

and hence `A=-2`, `C=2`

Subtracting the second equation from the fourth, we get:

`8B = 8`

and hence `B=1`, `D=0`

So:

`(x^2-16x+9)/(x^4+10x^2+9) = (-2x+1)/(x^2+1) + (2x)/(x^2+9)`