Problem 3 - Calculus Early Transcendentals

Question

What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following? a. A factor of \(x-3\) in the denominator b. A factor of \((x-4)^{3}\) in the denominator c. A factor of \(x^{2}+2 x+6\) in the denominator

Step-by-Step Solution

Answer

Question: Identify the terms that appear in the partial fraction decomposition of the given denominator factors: a. A factor of \(x-3\) in the denominator b. A factor of \((x-4)^{3}\) in the denominator c. A factor of \(x^{2}+2 x+6\) in the denominator Short Answer: a. \(\frac{A}{x-3}\) b. \(\frac{A}{x-4} + \frac{B}{(x-4)^2} + \frac{C}{(x-4)^3}\) c. \(\frac{Ax+B}{x^{2}+2x+6}\) Where A, B, and C are constants.

Step 1: a. A factor of \(x-3\) in the denominator

If the denominator has a factor of \((x-3)\), then the term(s) that should appear in the partial fraction decomposition is: $$\frac{A}{x-3}$$ where A is a constant.

Step 2: b. A factor of \((x-4)^{3}\) in the denominator

If the denominator has a factor of \((x-4)^{3}\), then the decomposition will have terms having powers 1 to 3, which means the term(s) that should appear in the partial fraction decomposition are: $$\frac{A}{x-4} + \frac{B}{(x-4)^2} + \frac{C}{(x-4)^3}$$ where A, B, and C are constants.

Step 3: c. A factor of \(x^{2}+2 x+6\) in the denominator

If the denominator has a factor of \(x^{2}+2x+6\), we notice first that this is an irreducible quadratic factor. Therefore, the term that should appear in the partial fraction decomposition is: $$\frac{Ax+B}{x^{2}+2x+6}$$ where A and B are constants.