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17Calculus Precalculus - Improper Fractions

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
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Rational Functions Containing Improper Fractions

One thing we haven't emphasized until now is that, to do partial fraction expansion, the highest power in the denominator must be greater than the highest power in the numerator. If this is not the case, then we have an improper fraction and partial fraction expansion will not work.

When we have an improper fraction, we need to do long division of polynomials to separate out the extra terms. When we have done that, we end up with a polynomial plus a proper fraction on which we can do partial fraction expansion.

As far as partial fraction expansion goes, nothing changes once you have a proper fraction.

Okay, so how do you determine if you have an improper fraction? Here is a video clip with a great explanation of this idea using 3 examples.

ExamSolutions - Algebraic Fractions : Improper fractions to mixed fractions (clip 1)

video by ExamSolutions

So as you saw in that clip, he determined that he had two improper fractions and one that was proper. So now that he has that information, he uses a numerical example in this next video clip to remind us of the general technique before working with the improper fractions. Have a look at this clip.

ExamSolutions - Algebraic Fractions : Improper fractions to mixed fractions (clip 2)

video by ExamSolutions

Okay, so now he uses this concept and long division to get what he calls mixed fractions on the two examples. Watch how he does it in this video clip.

ExamSolutions - Algebraic Fractions : Improper fractions to mixed fractions (clip 3)

video by ExamSolutions

So in that video clip, he ended up with 1 + a proper fraction in the first example and x - 3 + a proper fraction in the second example. That is where the video stops but, if the problem asked for him to do partial fraction expansion, he can do it on the proper fractions that he got. See how that works?

Before working the practice problems, you need to know the following techniques.
1. Long division of polynomials
2. All partial fractions techniques involving linear and quadratic factors, single and repeating.

Okay, let's work the practice problems.

Practice

Unless otherwise instructed, expand the given fraction using partial fraction expansion. Give your answer in exact terms.

\(\displaystyle{\frac{x^4+2x^3-4x^2-7x-6}{x^2-4}}\)

Problem Statement

Expand \(\displaystyle{\frac{x^4+2x^3-4x^2-7x-6}{x^2-4}}\) using partial fraction expansion. Give your answer in exact terms.

Solution

3093 video solution

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\(\displaystyle{ \frac{x^3+3}{x^2-2x-3} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^3+3}{x^2-2x-3} }\) using partial fraction expansion. Give your answer in exact terms.

Final Answer

\(\displaystyle{ \frac{x^3+3}{x^2-2x-3} }\) \(\displaystyle{ = x+2 + \frac{15/2}{x-3} - \frac{1/2}{x+1} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^3+3}{x^2-2x-3} }\) using partial fraction expansion. Give your answer in exact terms.

Solution

3100 video solution

Final Answer

\(\displaystyle{ \frac{x^3+3}{x^2-2x-3} }\) \(\displaystyle{ = x+2 + \frac{15/2}{x-3} - \frac{1/2}{x+1} }\)

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\(\displaystyle{ \frac{x^3+3x^2+6x+7}{x^2+3x+2} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^3+3x^2+6x+7}{x^2+3x+2} }\) using partial fraction expansion. Give your answer in exact terms.

Final Answer

\(\displaystyle{ \frac{x^3+3x^2+6x+7}{x^2+3x+2} }\) \(\displaystyle{ = x + \frac{1}{x+2} + \frac{3}{x+1} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^3+3x^2+6x+7}{x^2+3x+2} }\) using partial fraction expansion. Give your answer in exact terms.

Solution

3101 video solution

Final Answer

\(\displaystyle{ \frac{x^3+3x^2+6x+7}{x^2+3x+2} }\) \(\displaystyle{ = x + \frac{1}{x+2} + \frac{3}{x+1} }\)

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\(\displaystyle{ \frac{x^2+3x-5}{x^2-1} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^2+3x-5}{x^2-1} }\) using partial fraction expansion. Give your answer in exact terms.

Final Answer

\(\displaystyle{ \frac{x^2+3x-5}{x^2-1} }\) \(\displaystyle{ = 1 - \frac{1}{2(x-1)} + \frac{7}{2(x+1)} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^2+3x-5}{x^2-1} }\) using partial fraction expansion. Give your answer in exact terms.

Solution

3102 video solution

Final Answer

\(\displaystyle{ \frac{x^2+3x-5}{x^2-1} }\) \(\displaystyle{ = 1 - \frac{1}{2(x-1)} + \frac{7}{2(x+1)} }\)

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\(\displaystyle{ \frac{x^3-3x}{x^2-x-2} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^3-3x}{x^2-x-2} }\) using partial fraction expansion. Give your answer in exact terms.

Final Answer

\(\displaystyle{ \frac{x^3-3x}{x^2-x-2} }\) \(\displaystyle{ = x + 1 - \frac{2}{3(x+1)} + \frac{2}{3(x-2)} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^3-3x}{x^2-x-2} }\) using partial fraction expansion. Give your answer in exact terms.

Solution

3103 video solution

Final Answer

\(\displaystyle{ \frac{x^3-3x}{x^2-x-2} }\) \(\displaystyle{ = x + 1 - \frac{2}{3(x+1)} + \frac{2}{3(x-2)} }\)

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\(\displaystyle{ \frac{x^5-2x^4+x^3+x+5}{x^3-2x^2+x-2} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^5-2x^4+x^3+x+5}{x^3-2x^2+x-2} }\) using partial fraction expansion. Give your answer in exact terms.

Final Answer

\(\displaystyle{ \frac{x^5-2x^4+x^3+x+5}{x^3-2x^2+x-2} }\) \(\displaystyle{ = x^2 - \frac{x+1}{x^2+1} + \frac{3}{x-2} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^5-2x^4+x^3+x+5}{x^3-2x^2+x-2} }\) using partial fraction expansion. Give your answer in exact terms.

Solution

3104 video solution

Final Answer

\(\displaystyle{ \frac{x^5-2x^4+x^3+x+5}{x^3-2x^2+x-2} }\) \(\displaystyle{ = x^2 - \frac{x+1}{x^2+1} + \frac{3}{x-2} }\)

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\(\displaystyle{ \frac{15x-12x^2-1}{8x-6x^2-2} }\)

Problem Statement

Expand \(\displaystyle{ \frac{15x-12x^2-1}{8x-6x^2-2} }\) using partial fraction expansion. Give your answer in exact terms.

Final Answer

\(\displaystyle{ \frac{15x-12x^2-1}{8x-6x^2-2} }\) \(\displaystyle{ = 2 + \frac{1}{2(1-x)} + \frac{2}{3x-1} }\)

Problem Statement

Expand \(\displaystyle{ \frac{15x-12x^2-1}{8x-6x^2-2} }\) using partial fraction expansion. Give your answer in exact terms.

Solution

3105 video solution

Final Answer

\(\displaystyle{ \frac{15x-12x^2-1}{8x-6x^2-2} }\) \(\displaystyle{ = 2 + \frac{1}{2(1-x)} + \frac{2}{3x-1} }\)

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\(\displaystyle{ \frac{2x^2-3x-14}{x^2-x-2} }\)

Problem Statement

Expand \(\displaystyle{ \frac{2x^2-3x-14}{x^2-x-2} }\) using partial fraction expansion. Give your answer in exact terms.

Final Answer

\(\displaystyle{ \frac{2x^2-3x-14}{x^2-x-2} }\) \(\displaystyle{ = 2 + \frac{3}{x+1} - \frac{4}{x-2} }\)

Problem Statement

Expand \(\displaystyle{ \frac{2x^2-3x-14}{x^2-x-2} }\) using partial fraction expansion. Give your answer in exact terms.

Solution

3106 video solution

Final Answer

\(\displaystyle{ \frac{2x^2-3x-14}{x^2-x-2} }\) \(\displaystyle{ = 2 + \frac{3}{x+1} - \frac{4}{x-2} }\)

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\(\displaystyle{ \frac{(x+2)(x-2)}{(x+1)(x-1)} }\)

Problem Statement

Expand \(\displaystyle{ \frac{(x+2)(x-2)}{(x+1)(x-1)} }\) using partial fraction expansion. Give your answer in exact terms.

Final Answer

\(\displaystyle{ \frac{(x+2)(x-2)}{(x+1)(x-1)} }\) \(\displaystyle{ = 1 + \frac{3}{2(x+1)} - \frac{3}{2(x-1)} }\)

Problem Statement

Expand \(\displaystyle{ \frac{(x+2)(x-2)}{(x+1)(x-1)} }\) using partial fraction expansion. Give your answer in exact terms.

Solution

3107 video solution

Final Answer

\(\displaystyle{ \frac{(x+2)(x-2)}{(x+1)(x-1)} }\) \(\displaystyle{ = 1 + \frac{3}{2(x+1)} - \frac{3}{2(x-1)} }\)

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\(\displaystyle{ \frac{x^3-2x^2-25x+53}{25-x^2} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^3-2x^2-25x+53}{25-x^2} }\) using partial fraction expansion. Give your answer in exact terms.

Final Answer

\(\displaystyle{ \frac{x^3-2x^2-25x+53}{25-x^2} }\) \(\displaystyle{ = -x + 2 + \frac{3}{10(5+x)} + \frac{3}{10(5-x)} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^3-2x^2-25x+53}{25-x^2} }\) using partial fraction expansion. Give your answer in exact terms.

Solution

3108 video solution

Final Answer

\(\displaystyle{ \frac{x^3-2x^2-25x+53}{25-x^2} }\) \(\displaystyle{ = -x + 2 + \frac{3}{10(5+x)} + \frac{3}{10(5-x)} }\)

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\(\displaystyle{ \frac{x^3-x^2-5x-7}{x^2-2x-3} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^3-x^2-5x-7}{x^2-2x-3} }\) using partial fraction expansion. Give your answer in exact terms.

Final Answer

\(\displaystyle{ \frac{x^3-x^2-5x-7}{x^2-2x-3} }\) \(\displaystyle{ x + 1 - \frac{1}{x+1} + \frac{1}{x-3} }\)

Problem Statement

Expand \(\displaystyle{ \frac{x^3-x^2-5x-7}{x^2-2x-3} }\) using partial fraction expansion. Give your answer in exact terms.

Solution

3109 video solution

Final Answer

\(\displaystyle{ \frac{x^3-x^2-5x-7}{x^2-2x-3} }\) \(\displaystyle{ x + 1 - \frac{1}{x+1} + \frac{1}{x-3} }\)

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Practice Instructions

Unless otherwise instructed, expand the given fraction using partial fraction expansion. Give your answer in exact terms.

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