What is the partial fraction decomposition of 7x^2-6x+9/3x(4x^2+9)

Simplify the expression: x+7x2+3x+2=x+7(x+1)(x+2)

The form of the partial fraction decomposition is

x+7(x+1)(x+2)=Ax+1+Bx+2

Write the right-hand side as a single fraction:

x+7(x+1)(x+2)=(x+1)B+(x+2)A(x+1)(x+2)

The denominators are equal, so we require the equality of the numerators:

x+7=(x+1)B+(x+2)A

Expand the right-hand side:

x+7=xA+xB+2A+B

Collect up the like terms:

x+7=x(A+B)+2A+B

The coefficients near the like terms should be equal, so the following system is obtained:

{A+B=12A+B=7

Solving it (for steps, see system of equations calculator), we get that A=6, B=−5

Therefore,

x+7(x+1)(x+2)=6x+1+−5x+2

x + 7x\textasciicircum{2} + 3 x + 2=6x+1+−5x+2