Problem 15 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{4}{2 x^{2}-5 x-3}$$

Step-by-Step Solution

Answer

\(\frac{4}{2x^{2}-5x-3} = \frac{2}{2x+1} + \frac{1}{x-3}\)

Step 1: Factorize the denominator

Re-write the denominator quadratic \(2x^{2}-5x-3\) as \((2x+1)(x-3)\). This is achieved by finding two numbers that multiply to give \(2*-3=-6\) and at the same time adds up to give \(-5\). So, performing the factorization, the denominator stands: \(2x^{2}-5x-3 = (2x+1)(x-3)\)

Step 2: Write the partial fraction decomposition

Now, express the fraction as a sum of partial fractions: \[\frac{4}{2x^{2}-5x-3} = \frac{A}{2x+1} + \frac{B}{x-3}\] where A and B are constants that need to be found.

Step 3: Find the constants A and B

To find the values of A and B, first, equate both sides of the equation to the common denominator \((2x+1)(x-3)\) and simplify. This result: \[4 = A(x-3) + B(2x+1)\]. Then, using a system of equations to find the values of A and B, you can let \(x=3\) which gives \(A=2\), and then \(x=-1/2\) which gives \(B=1\).

Step 4: Write the final partial fraction decomposition

Finally, substitute A=2 and B=1 into the partial fraction decomposition: \[\frac{4}{2x^{2}-5x-3} = \frac{2}{2x+1} + \frac{1}{x-3}\]