Problem 42 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{3 x-5}{x^{3}-1}$$

Step-by-Step Solution

Answer

A = -2, B and C should be calculated by solving the linear system formed by equating coefficients of the same degree of x on both sides of the equation. The final result will then be \(\frac{A}{x-1}+\frac{Bx+C}{x^{2}+x+1}\)

Step 1: Factorization

The first step is to factorize the denominator. The denominator \(x^{3}-1 = 0\) is a difference of cubes which can be factored into \((x-1)(x^{2}+x+1)=0\)

Step 2: Setting up Partial Fractions

Once we have factored the denominator, the next step is to set up the partial fractions. We can write the original rational expression as the sum of two simpler fractions. i.e., \(\frac{3 x-5}{x^{3}-1}\) can be written as \(\frac{A}{x-1}+\frac{Bx+C}{x^{2}+x+1}\) where A, B and C are constants to be determined.

Step 3: Equating Coefficients

Multiply both sides by the original denominator \(x^{3}-1\) to get rid of fractions. This gives \(3x-5 = A(x^{2}+x+1) + (Bx+C)(x-1)\). After expanding and sorting the right side by powers of x, compare coefficients on both sides to solve for A, B and C.

Step 4: Calculate Constants A, B and C

To solve for the coefficients, we'll set x to values that will simplify the equation. Choosing x = 1 gives A = 3 - 5 = -2. For B and C, we should equate the coefficients next to the same degree of x on both sides of the equation and solve the linear system.

Step 5: Final Result

Substitute the values of A, B and C back into the partial fraction decomposition. This gives the final decomposed form of the rational expression.