Problem 34 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{10 x^{2}+2 x}{(x-1)^{2}\left(x^{2}+2\right)}$$

Step-by-Step Solution

Answer

\(\frac{10x^2+2x}{(x-1)^2(x^2+2)} = \frac{1}{x-1} + \frac{6}{(x-1)^2} + \frac{-8x-4}{x^2+2}\)

Step 1: Setup Partial Fractions

First, let's break the given fraction into the sum of partial fractions. We can write it in the following form: \(\frac{10x^2+2x}{(x-1)^2(x^2+2)} = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{Cx+D}{x^2+2}\). A, B, C, and D are constants which we need to find. The first two terms are due to the linear term \((x-1)^2\) and the third term is due to the quadratic term \(x^2+2\).

Step 2: Clear the Fraction

Now, to find the constants A, B, C, and D, we will clear the fraction and group the like terms together. To do this, we can multiply both sides of the equation by the denominator \((x-1)^2(x^2+2)\) of the left side. We get, \(10x^2+2x = A(x-1)(x^2+2) + B(x^2+2) + (Cx+D)(x-1)^2\).

Step 3: Evaluate constants

Now we will choose specific values for x that will simplify the equations and solve for A, B, C, and D. Let's start with x = 1, then the equation will be only in terms of B, \(12 = 2B\), solving this we get B = 6. Next by differentiating the equation with respect to x and then substituting x=1, will cancel A giving us \(C = -8\). Now to find A and D substitute x=i, we can solve the system of equations giving us A = 1 and D = -4.

Step 4: Write the final answer

Now that we have the values of A, B, C, and D, substitute them back into the partial fractions from step 1. The partial fraction decomposition of the given expression is equal to \(\frac{1}{x-1} + \frac{6}{(x-1)^2} + \frac{-8x-4}{x^2+2}\).