Partial Fractions

This chapter explores partial fractions. The objectives for this chapter are to be able to express a fraction with linear fractions in the denominator as partial fractions and expressing a fraction with repeated linear factors in the denominator as partial fractions. Before attempting the chapter you must have prior knowledge of simplifying and adding algebraic fractions and understanding algebraic fractions and understanding lowest common denominators for algebraic fractions including repeated factors in the denominators.

What is a partial fraction?

Remember when simplifying algebraic fractions such as shown below
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…we simplify by writing them over a common denominator. The common denominator is;
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…so we write…
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Above we have multiplied the first fraction through by (x – 2) and the second by (x + 1)/ Now we can simplify;
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That leads to conclusion that;
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The partial fractions are;
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We do the process backwards; it is about how we split the final fraction apart to give use the two starting partial fractions.

Types of partial fraction problem

There are four groups of problems that we can split to form the type of partial fractions.

Type 1

This is where the denominator is of the form (ax + b)(cx + d). This is the simple one;
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Type 2

This is where the denominator has a repeated factor and is in the form of (ax + b)(cx + d)² for example;
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Type 3

This is where the denominator has one quadratic factor and in the form (ax + b)(cx² + d) for example;
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Type 4

This type of partial fraction problem is where the highest power of the numerator is equal to or larger than the highest power of the denominator such as that shown below;
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In the above example 3 is the largest power of the numerator and 2 is the largest power of the denominator.
Below we shall explore each type separately.

Type 1: Denominator of the form (ax + b)(cx + d)

Below is an example of this partial fraction problem.
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We can rewrite it in the partial fractions form
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We just need to find A and B we add the two fractions naturally.
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That must mean that;
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…we write the above as;
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…we take the numerators as shown below.
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To find B we must have the A part disappear, to do this we let x = 3. Then we have;
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…we simplify;
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So now we know that;
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Next we let x=-2 to get rid of B or to make the B term zero.
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…we simplify;
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…so now we know that;
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So the partial fractions are;
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We can double check it by working it out;
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Below are more examples of how to split into partial fractions.

Example

Split the following into partial fractions
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First we factorise the denominators so that it is in the form of (ax + b)(cx + d)
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Now we need to split the fractions into the form;
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We add the form naturally…
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Now we can equate the numerators as shown below;
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…we let x=-2
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…we simplify to get…
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Now we know that;
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Next we let x=1/2 as shown below;
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…we simplify to get;
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Now we know that…
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The partial fractions we form are;
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or
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Example

Here is another example; Split the following into partial fractions
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Notice that this time there are 3 linear fractions on the bottom so we setup as 3 partial fractions.
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We can substitute in and add naturally.
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Now we can equate the top lines;
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We can eliminate B and C first to find A first by letting x=0.
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Now we know that;
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To find B we let x=-2 to eliminate A and C.
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Now we know that;
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Next to eliminate A and B to find C we let x=5.
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…we know that;
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So the partial fractions are;
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Type 2: Denominators of the form (ax + b)(cx + d)²

This type of partial fractions problem has repeated factor on the bottom line. This would be written in 3 partial fractions as shown below.
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We can add the fractions naturally, the common denominator is;
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…that is;
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Now we can equate the top lines;
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…we let x = 1
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Now we know that;
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Next we let x=-2
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Now we know that;
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This time to find C we can’t just let x be a number that eliminates A and B. We need to equate the x² terms.
The side circled below has just 1 lot of x²
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So we have…
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In the part circled below multiplied out we get Ax²;
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so we have…
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From the term circled below we get ox²
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…so we have
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From the term circled below we get cx²
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…so we have
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Now we know that;
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So the partial fractions are;
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Example

Split the following into partial fractions.
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First we factorise the bottom line as shown below. First we take a common factor of x.
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…and then we factorise further into brackets;
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We will need to setup into partial fractions into the form.
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…we add up the fractions naturally to get;
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Next equate the top lines as shown below.
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First we let x=-5
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Now we know that;
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Next we let x=0
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Now we know that;
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Next we equate the x² terms;
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Now we know that;
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So the partial fractions are;
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