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

1 Answer
Sep 23, 2017

(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)