Partial Fraction Decomposition: Rules & Examples

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  • 0:04 Partial Fraction Decomposition
  • 1:11 Setting Up the Work
  • 11:45 Guidelines
  • 13:40 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

What if you had a way to expand certain large math expressions into smaller pieces? This would make some calculus integrals easier to solve. In this lesson, we explore such a method: partial fraction decomposition.

Partial Fraction Decomposition

Building a house of cards is getting complicated results from something simpler, which is the opposite of partial fraction decomposition (PFD), where you get simpler results from something complicated. But just like a house of cards, partial fraction decomposition needs structure and rules.

Partial Fraction Decomposition is just like a house of cards.
A House of Cards

I'm sure you've seen fractions before, but what about polynomial fractions? These have polynomials for both the numerator and denominator. For example:

polynomial fraction

Both polynomials are in standard form: the terms are ordered from the highest exponent to the lowest. Here's another key idea: the order of a polynomial, which is the highest numbered exponent. Our standard form numerator polynomial has a first term of 2x1. The numerator order is 1. The order of the denominator polynomial is 2, since x2 is the highest exponent. The first rule of PFD is the denominator order must be greater than the numerator order.

Setting up the Work

Please don't panic with the following equation. This is just four cases of PFD combined into a single expression.

four cases of PFD

The (x + 2) factor is linear, while (x2 - 2x + 2) is quadratic. The (x - 1)2 is a linear factor raised to a power, while (x2 + 4)2 is a quadratic factor raised to a power. The arrows point to the PFD.

Linear Factors

Linear factors have x raised to the first power:

examples of linear factors

Here is a fraction with two linear factors:

polynomial fraction with 2 linear factors

Remember the importance of structure in the house of cards? Our structure here is:

structure of the decomposition for linear factors

Like a house of cards, isn't it? Okay, maybe not, but we have a rule for linear factors: for each linear factor, write a new fraction of a capital letter over the linear factor. Then add these new fractions together.

In this example, the right-hand side of the equation becomes the sum of two new fractions. This is how we find a common denominator:

common denominator for linear factors

Since the left-hand side must equal the right-hand side, the numerators must equal each other, and we can simplify our equation:

equating numerators

Any value may be substituted for x, although some values will simplify better than others. Letting x = 1 wipes out the A since 1 - 1 = 0, and with some algebra gives us B = 1. Letting x = -2, we can wipe out the B with -2 + 2 = 0, gives us A = 3. We now have our decomposition:

the resulting decomposition for the linear factors example

Quadratic factors have x raised to the second power:

examples of quadratic factors

Let's decompose this equation:

example with a quadratic factor

Quadratic Factors

The first factor in the denominator is quadratic. Here's the rule for quadratic factors: write Ax + B in the numerator of a new fraction and the quadratic factor as the denominator.

The second factor in the denominator is linear. In our new fraction, it gets a single letter over the linear factor. Our structure here is:

structure for PFD when dealing with a quadratic factor

As with the linear equations, we use common denominator and then equate numerators:

equating numerators for the quadratic factor example

Substituting x = -2 (to wipe out A and B), gives C = 3. Expanding the right-hand side multiplications and grouping terms:

expansion of the RHS and grouping terms

What if we compare the left-hand side with the right-hand side? To be clear, we are looking at:

the LHS and RHS of the quadratic factor example

On the right-hand side, look at what is multiplying the x2 - it is A + C. Now we look at the left-hand side and see the x 2 term being multiplied by 5. Our conclusion? A + C = 5.

The 8 all by itself on the left-hand side compares to what on the right-hand side? Here's a hint: on the right-hand side, find the terms not being multiplied by an x or an x2. You are correct! 2B + 2C = 8.

Having made these comparisons, we can write:

conclusions after comparing RHS with LHS

Using C = 3 gives us A = 2 and B = 1. The result is:

final result for the quadratic factor example

Linear Factors Raised to a Power

What about a linear factor raised to the second power?

a PDF with a linear factor raised to a second power

The structure here is:

the structure of a PFD with a linear factor rasied to a second power

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