Find the partial decomposition
11x - 22/(2x - 1)(x - 6)
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)