The rational function (x-4)/x(2x-1) can be expressed as the sum of two partial fractions: A/x and B/(2x-1). Find the value of A – B.
a) -11
b) 11
c) -3
d) 3
Let (x-4)/(x(2x-1)) = A/x + B/(2x-1) = [ A(2x-1) + Bx]/(x(2x-1))
(how nice of them to factor the denominator for us)
x - 4 = A(2x-1) + Bx
This becomes an identity, thus true for all values of x
let x = 0 -----> -4 = -A or A = 4
let x = 1/2 ----> -7/2 = B/2 or B = -7
check: 4/x - 7/(2x-1) = (4(2x-1) - 7x)/(x(2x-1)) = (x - 4)/(x(2x-1))
so they want B-A = ....
a/x + b/(2x-1) = (x-4) / [ x(2x-1) ] I assume you mean
multiply both sides by [x(2x-1)]
a(2x-1) + b x = x-4
2ax - a + bx = 1x - 4
(2a+b)x = 1x
-a = -4
so a = 4
(8+b ) = 1
b = -7
4 - (-7) = 11
To find the value of A - B, we need to first find the values of A and B separately.
The given rational function is (x - 4)/(x(2x - 1)). By factoring the denominator, we have x(2x - 1) = x * (2x - 1).
Now we can express this rational function as the sum of two partial fractions:
(x - 4)/(x(2x - 1)) = A/x + B/(2x - 1)
To determine the values of A and B, we can use a common approach known as the method of partial fractions. Here's how it works:
1. Clear the denominator: Multiply the entire equation by the common denominator (x(2x - 1)) to eliminate the fractions:
(x - 4) = A * (2x - 1) + B * x
2. Solve for x:
Expand the equation by distributing the terms:
x - 4 = 2Ax - A + Bx
Combine like terms:
x - 4 = (2A + B)x - A
The coefficients of 'x' on both sides of the equation must be equal, as well as the constant terms:
2A + B = 1 (equation 1)
-A = -4 (equation 2)
From equation 2, we can determine that A = 4.
Substituting A = 4 into equation 1:
2(4) + B = 1
8 + B = 1
B = -7
Now we have the values of A (4) and B (-7). To find A - B, subtract the second value from the first:
A - B = 4 - (-7) = 4 + 7 = 11
Therefore, the value of A - B is 11. Thus, the correct option is b) 11.