Exercises
Challenge your understanding of partial fraction decomposition for rational functions. This quiz covers when polynomial division is required, decompositions with distinct and repeated linear factors, irreducible quadratic factors, and the correct numerator forms for each term. Practice finding unknown constants, rewriting improper rational expressions, and identifying decompositions involving factors such as x, x − 1, and quadratic polynomials. You will also apply decomposed forms to recognize features such as horizontal asymptotes. Ideal for algebra and precalculus students strengthening their rational-expression skills.
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Partial fraction decomposition begins with a proper rational expression, whose numerator has a lower degree than its denominator. Otherwise, polynomial division must be performed first.
Writing the expression as A/(x − 1) + B/(x + 2) gives A = 5/3 and B = −5/3. Combining these fractions reproduces the original numerator of 5.
A repeated linear factor requires one term for each power up to its multiplicity. Thus, both x − 3 and (x − 3)² need terms, along with a term for x + 1.
The numerator above an irreducible quadratic must have degree less than two. Therefore, its general form is the linear expression Bx + C.
Polynomial division gives x² + 1 = (x − 1)(x + 1) + 2. Dividing by x − 1 produces x + 1 + 2/(x − 1).
Combining 1/x − 1/(x + 1) gives ((x + 1) − x)/(x(x + 1)), which simplifies to 1/(x(x + 1)).
After clearing denominators, set x = 2. This gives 7 = 3A, so A = 7/3. This substitution isolates the coefficient associated with x − 2.
Clearing denominators gives x + 2 = A(x − 1) + B. Setting x = 1 eliminates the A term and yields B = 3.
Since x² − 1 = (x − 1)(x + 1), use A/(x − 1) + B/(x + 1). Solving gives A = B = 1/2, producing the stated decomposition.
As |x| grows, 2/(x − 1) approaches zero, so f(x) approaches 3. Therefore, the horizontal asymptote is y = 3.
Clearing denominators gives x² + 2 = A(x² + 1) + x(Bx + C). Comparing constant terms gives A = 2.
Using the roots 0, 1, and −1 to isolate coefficients gives −1, 1/2, and 1/2, respectively. Substituting these values yields the correct decomposition.
Use A/(x − 2) + (Bx + C)/(x² + 4x + 5). Equating coefficients gives A = 3/17, B = −3/17, and C = −1/17, which matches the second expression.
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