Problem 36 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{3 x^{2}-2 x+8}{x^{3}+2 x^{2}+4 x+8}$$

Step-by-Step Solution

Answer

Since the denominator \(x^{3}+2x^{2}+4x+8\) can't be factored and the expression is already a proper rational function, the partial fraction decomposition is simply the function itself: \(\frac{3 x^{2}-2 x+8}{x^{3}+2 x^{2}+4 x+8}\).

Step 1: Factorization of the Denominator

First, factor the polynomial in the denominator. In this case, the denominator is \(x^{3}+2x^{2}+4x+8\). However, this polynomial can't be factored using rational numbers. Therefore, it is already in its simplest form.

Step 2: Formation of Partial Fractions

The given function is a proper rational function (degree of the numerator is less than the degree of the denominator). So, the expression can directly be written as the sum of simpler fractions (partial fractions). Therefore, it's already a partial fraction in itself.