How do you express (5x - 4 ) / (x^2 -4x) 5x4x24x in partial fractions?

1 Answer
Oct 22, 2016

(5x - 4)/(x^2 - 4x) = 4/(x - 4) + 1/x5x4x24x=4x4+1x

Explanation:

The denominator can be written as x(x - 4)x(x4).

We can rewrite this as follows:

A/(x - 4) + B/x = (5x- 4)/(x^2 - 4x)Ax4+Bx=5x4x24x

Put on a common denominator.

A(x) + B(x - 4) = 5x - 4A(x)+B(x4)=5x4

Ax + Bx - 4B = 5x - 4Ax+Bx4B=5x4

(A + B)x - 4B = 5x - 4(A+B)x4B=5x4

We can hence write the following system of equations.

A + B = 5A+B=5
-4B = -44B=4

Solving, we have that B = 1B=1 and that A = 4A=4

We can now reinsert into the original equation to get our partial fraction decomposition.

4/(x- 4) + 1/x = (5x - 4)/(x^2 - 4x)4x4+1x=5x4x24x

Let's quickly check the validity of our answer.

What is the sum of 4/(x - 4) + 1/x4x4+1x?

Put on a common denominator.

(4(x))/((x - 4)(x)) + (1(x- 4))/(x(x - 4)) = (4x + x - 4)/(x(x - 4)) = (5x - 4)/(x^2 - 4x)4(x)(x4)(x)+1(x4)x(x4)=4x+x4x(x4)=5x4x24x

This verifies the initial expression, so we have done our decomposition properly.

Hopefully this helps!