How do you write the partial fraction decomposition of the rational expression X25x+6X3X2+2X?

1 Answer
Dec 30, 2015

3x2(x+1)x2x+2

Explanation:

Factor out an x in the denominator

x25x+6x(x2x+2)=Ax+Bx+Cx2x+2

Multiply both sides by x(x2x+2)

x25x+6=A(x2x+2)+x(Bx+C)

Distribute on right hand side

x25x+6=Ax2Ax+2A+Bx2+Cx

Factor right hand side

x25x+6=x2(A+B)+x(A+C)+2A

Equate coefficients on the left hand side with the right

2A=6

A+C=5

A+B=1

Solving we get

A=3

B=2

C=2

Putting it all together

x25x+6x3x2+2x=3x+2x2x2x+2

Pull out negative in 2x2

x25x+6x3x2+2x=3x2x+2x2x+2

Factor out a 2 in 2x+2

x25x+6x3x2+2x=3x2(x+1)x2x+2