In partial fraction decomposition, to find a constant a in a term a/(...) you typically do what?

In partial fraction decomposition, to find a constant a in a term a/(...) you typically do what?

• A. Set the bottom factor to zero to solve for a
• B. Differentiate both sides
• C. Integrate both sides
• D. Multiply through by the original denominator

Answer: (A)

The technique here is the cover-up method in partial fractions: to find the coefficient a in a term a/(x − r) times the rest, you clear the fractions by multiplying through by the full denominator and then substitute the value of x that makes the bottom factor zero. This substitution cancels all the other partial-fraction terms, since they all contain that same factor in the denominator, leaving you with a single simple equation that gives a directly. In other words, setting the bottom factor to zero isolates the coefficient you’re after. Differentiating or integrating doesn’t isolate a in this setup, and while multiplying through by the original denominator is part of the procedure, you still need that substitution to actually solve for a. For simple linear factors, this substitution is the standard, quickest way to find the coefficient.

The technique here is the cover-up method in partial fractions: to find the coefficient a in a term a/(x − r) times the rest, you clear the fractions by multiplying through by the full denominator and then substitute the value of x that makes the bottom factor zero. This substitution cancels all the other partial-fraction terms, since they all contain that same factor in the denominator, leaving you with a single simple equation that gives a directly. In other words, setting the bottom factor to zero isolates the coefficient you’re after. Differentiating or integrating doesn’t isolate a in this setup, and while multiplying through by the original denominator is part of the procedure, you still need that substitution to actually solve for a. For simple linear factors, this substitution is the standard, quickest way to find the coefficient.