Find the partial decomposition

Question

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

Answer 1

To find the partial decomposition of the expression, we first need to factor the denominator:

(2x - 1)(x - 6)

Next, we need to express the numerator as a sum of two fractions with the same denominator:

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

Now, we need to solve for A and B by equating the coefficients of like terms on both sides of the equation:

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

Now, we equate the coefficients of x terms and constant terms on both sides:

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

Now, we solve these simultaneous equations to find the values of A and B. Multiplying the first equation by 6 and adding it to the second equation:

66 = 12A + 6B
-22 = -A - 6B
--------------------
44 = 11A
A = 4

Now, substitute the value of A back into the first equation to solve for B:

11 = 2(4) + B
11 = 8 + B
B = 3

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

4/(2x - 1) + 3/(x - 6)