How do you express (x^2-16x+9)/(x^4+10x^2+9)x2−16x+9x4+10x2+9 in partial fractions?
1 Answer
Explanation:
(x^2-16x+9)/(x^4+10x^2+9) = (x^2-16x+9)/((x^2+1)(x^2+9))x2−16x+9x4+10x2+9=x2−16x+9(x2+1)(x2+9)
color(white)((x^2-16x+9)/(x^4+10x^2+9)) = (Ax+B)/(x^2+1) + (Cx+D)/(x^2+9)x2−16x+9x4+10x2+9=Ax+Bx2+1+Cx+Dx2+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)x2−16x+9x4+10x2+9=(Ax+B)(x2+9)+(Cx+D)(x2+1)x4+10x2+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)x2−16x+9x4+10x2+9=(A+C)x3+(B+D)x2+(9A+C)x+(9B+D)x4+10x2+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
Subtracting the second equation from the fourth, we get:
8B = 8
and hence
So:
(x^2-16x+9)/(x^4+10x^2+9) = (-2x+1)/(x^2+1) + (2x)/(x^2+9)
