Which statement best describes partial fraction decomposition of a rational function with linear factors?

Which statement best describes partial fraction decomposition of a rational function with linear factors?

• A. It expresses the function as a sum of terms with denominators corresponding to those factors
• B. It differentiates the function term-by-term
• C. It multiplies the function by its conjugate
• D. It integrates the function term-by-term

Answer: (A)

Partial fraction decomposition rewrites a rational function whose denominator factors into linear terms as a sum of simpler fractions, each with one of those linear factors in the denominator. This means you express the function as something like A/(x − r1) plus B/(x − r2) plus ... (and, if a factor repeats, you include corresponding higher-power terms). The constants are found by clearing the denominators and equating coefficients, which makes the original expression easier to work with, especially for integration since each term is simple to integrate. This is why that description—expressing the function as a sum of terms with denominators corresponding to those factors—best captures what partial fraction decomposition does. Differentiating term-by-term, multiplying by a conjugate, or integrating term-by-term describe other techniques or steps, not the decomposition itself.

Partial fraction decomposition rewrites a rational function whose denominator factors into linear terms as a sum of simpler fractions, each with one of those linear factors in the denominator. This means you express the function as something like A/(x − r1) plus B/(x − r2) plus ... (and, if a factor repeats, you include corresponding higher-power terms). The constants are found by clearing the denominators and equating coefficients, which makes the original expression easier to work with, especially for integration since each term is simple to integrate. This is why that description—expressing the function as a sum of terms with denominators corresponding to those factors—best captures what partial fraction decomposition does. Differentiating term-by-term, multiplying by a conjugate, or integrating term-by-term describe other techniques or steps, not the decomposition itself.