Find the partial decomposition

Question

11x - 22/(2x - 1)(x - 6)

Answer 1

To find the partial decomposition of the rational function 11x - 22/(2x - 1)(x - 6), we need to write it in the form of A/(2x - 1) + B/(x - 6).

So, let's first write it as a single fraction:

11x - 22/(2x - 1)(x - 6) = A/(2x - 1) + B/(x - 6)

Multiplying both sides by (2x - 1)(x - 6), we get:

11x - 22 = A(x - 6) + B(2x - 1)

Expanding the right side:

11x - 22 = Ax - 6A + 2Bx - B

Collecting like terms:

11x - 22 = (A + 2B)x + (-6A - B)

Now we can compare coefficients:

Coefficients of x on both sides should be equal: A + 2B = 11
Constant terms on both sides should be equal: -6A - B = -22

Now, we solve for A and B by solving the system of equations:

From the first equation, let's solve for A:
A = 11 - 2B

Substitute A = 11 - 2B into the second equation:

-6(11 - 2B) - B = -22
-66 + 12B - B = -22
-67 + 11B = -22
11B = 45
B = 45/11

Now, substitute B = 45/11 back into A = 11 - 2B to find A:

A = 11 - 2(45/11)
A = 11 - 90/11
A = (121 - 90)/11
A = 31/11

Therefore, the partial decomposition of 11x - 22/(2x - 1)(x - 6) is:

31/11/(2x - 1) + 45/11/(x - 6)