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Cogito

AP Calculus BC · Chapter 6 · Lesson 2

Practice Worksheet

Partial Fractions

Splitting a rational function into pieces.

12 problems · about 22 minutes · FUN-6.F

NameDate

Figure — use these to answer the problems

Each partial fraction piece behaves like this curve, and each integrates to a logarithm.
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y = 1/x + 0
  • Point(2, 0.50)
a1
Point on the curve2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. A denominator with 4 distinct linear factors. How many partial fractions?

    Answer

  2. What must be checked before decomposing?

    • That the numerator degree is lower than the denominator.
    • That there are exactly two factors.

  3. A denominator with 2 distinct linear factors. How many partial fractions?

    Answer

  4. A denominator with 3 distinct linear factors. How many partial fractions?

    Answer

  5. A factor repeated twice. How many terms does it need?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Numerator degree 3, denominator degree 2. Divide first? 1 yes, 0 no.

    Answer

  2. Each simple partial fraction integrates to what? 1 a logarithm, 2 a polynomial.

    Answer

  3. x² − 4 factors into how many linear factors?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Put the partial fractions process in order.

    Write 1 to 4 in the boxes to put these in order.

    • Check the fraction is proper, dividing first if not.
    • Factor the denominator.
    • Write one term per factor with unknown constants.
    • Substitute the roots to find each constant.

  2. A factor repeated three times. How many terms?

    Answer

  3. The Constant

    For 1 = A(x − 2) + B(x − 1), substituting x = 2 gives B. What is B?

    Answer

  4. The Other One

    Substituting x = 1 in the same equation gives A. What is A?

    Answer