Problem 19 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{4 x^{2}-7 x-3}{x^{3}-x}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \( \frac{4x^{2}-7x-3}{x(x-1)(x+1)} \) is : \(\frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+1}\) where A, B, and C are real numbers obtained by solving the system of equations from Step 5.

Step 1: Factorize the Denominator

Firstly, factorize the denominator. The cubic polynomial can be factored as \(x^{3}-x = x(x^2-1)= x(x-1)(x+1)\)

Step 2: Set up the Partial Fractions

Next, set up the partial fractions with unknown coefficients. This can be written as: \(\frac{4x^{2}-7x-3}{x(x-1)(x+1)}= \frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+1}\)

Step 3: Clear the fractions

Multiply through by the denominator \(x(x-1)(x+1)\) to get rid of the fractions. So, you get: \(4x^{2}-7x-3 = A(x-1)(x+1) + Bx(x+1) +C x(x-1)\)

Step 4: Expansion and Simplification

Expand the right hand side and collect like terms. This will result in a polynomial of the same degree as the left hand side.

Step 5: Comparing Coefficients

Set coefficients for terms in the resulted polynomial equal to the coefficients for the same power of x in the original polynomial. This sets up a system of linear equations.

Step 6: Solve for A, B and C

Solve this system of linear equations for A, B and C. These will be the coefficients (numerators) of the partial fractions.

Step 7: Write the Decomposed Fractions

Write the decomposed fractions by replacing A, B and C in the set-up from Step 2 with the values obtained in Step 6.