please help me solve this partial-fraction decomposition!
12x-57/(x-3)(x-10)
let
A/(x-3) + B(x-10) = (12x-57)/((x-3)(x-10))
then
A(x-10) + B(x-3)/((x-3)(x-10)) = (12x-57)/((x-3)(x-10) )
or
A(x-10) + B(x-3) = 12x - 57
let x = 10
B(7) = 63 ----> B = 9
let x = 3
A(-7) = -21 ----> A = 3
so:
(12x-57)/((x-10)(x-3)) = 3/(x-3) + 9/(x-10)
thank you! :)
To solve the partial fraction decomposition of the expression (12x-57) / ((x-3)(x-10)), follow these steps:
Step 1: Factorize the denominator.
The denominator of the fraction is (x-3)(x-10). To simplify further, we need to factorize it completely.
(x - 3)(x - 10) is already in factored form, so no further factorization is needed.
Step 2: Write the partial fraction decomposition.
The general form of a partial fraction decomposition is A/(x - 3) + B/(x - 10), where A and B are unknown constants.
Step 3: Determine the values of A and B.
To find the values of A and B, we multiply both sides of the equation by the least common denominator (LCD), which is (x - 3)(x - 10). This step will eliminate the denominators on the left side of the equation.
(12x - 57) = A(x - 10) + B(x - 3)
Step 4: Solve for A and B.
To solve for A and B, we need to choose appropriate values of x that will simplify the equation.
We can simplify the equation by plugging in values for x that will cancel out one of the terms.
Let's choose x = 3.
(12(3) - 57) = A(3 - 10) + B(3 - 3)
(-3) = -7A
Now, let's choose x = 10.
(12(10) - 57) = A(10 - 10) + B(10 - 3)
(123) = 7B
We now have two equations:
-3 = -7A
123 = 7B
Solving these equations, we find that A = 3/7 and B = 123/7.
Step 5: Write the final partial fraction decomposition.
Now that we have found the values of A and B, we can write the final partial fraction decomposition:
(12x - 57) / ((x - 3)(x - 10)) = (3/7) / (x - 3) + (123/7) / (x - 10)