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Decompose the rational expression (5x + 3) / ((2x - 1)(x + 2)) into partial fractions.
ScanSolve AI Answer
Step-by-step solution
5 stepsThis problem involves partial fraction decomposition, which is a method to break down a complex rational expression into simpler fractions.
Set up the partial fraction decomposition*.
The denominator has two distinct linear factors: (2x - 1) and (x + 2). Therefore, we can write the expression as a sum of two fractions with these factors as denominators and unknown constants (A and B) as numerators.
(5x + 3) / ((2x - 1)(x + 2)) = A/(2x - 1) + B/(x + 2)
Combine the partial fractions*.
To solve for A and B, we first find a common denominator for the right side, which is (2x - 1)(x + 2).
A/(2x - 1) + B/(x + 2) = (A(x + 2) + B(2x - 1)) / ((2x - 1)(x + 2))
Equate the numerators*.
Since the denominators are now the same, the numerators must be equal.
5x + 3 = A(x + 2) + B(2x - 1)
Solve for A and B*.
We can find A and B by choosing specific values for x that simplify the equation.
To find A, let 2x - 1 = 0.
This means x = 1/2. Substitute x = 1/2 into the equation:5(1/2) + 3 = A(1/2 + 2) + B(2(1/2) - 1)
5/2 + 6/2 = A(1/2 + 4/2) + B(1 - 1)
11/2 = A(5/2) + B(0)
11/2 = 5A/2
Multiply both sides by 2:
11 = 5A
A = 11/5To find B, let x + 2 = 0.
This means x = -2. Substitute x = -2 into the equation:5(-2) + 3 = A(-2 + 2) + B(2(-2) - 1)
-10 + 3 = A(0) + B(-4 - 1)
-7 = B(-5)
-7 = -5B
B = (-7)/(-5)
B = 7/5Write the final partial fraction decomposition*.
Substitute the values of A and B back into the setup from Step 1.
(5x + 3) / ((2x - 1)(x + 2)) = (11/5)/(2x - 1) + (7/5)/(x + 2)
This can also be written as:
= 11 / (5(2x - 1)) + 7 / (5(x + 2))
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