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.
I’m sure you’ve seen fractions before, but what about polynomial fractions? These have polynomials for both the numerator and denominator.
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.
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 have x raised to the first power:
Here is a fraction with two linear factors:
Remember the importance of structure in the house of cards? Our structure here is:
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:
Since the left-hand side must equal the right-hand side, the numerators must equal each other, and we can simplify our equation:
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:
Quadratic factors have x raised to the second power:
Let’s decompose this equation:
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:
As with the linear equations, we use common denominator and then equate numerators:
Substituting x = -2 (to wipe out A and B), gives C = 3.
Expanding the right-hand side multiplications and grouping terms:
What if we compare the left-hand side with the right-hand side? To be clear, we are looking at:
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:
Using C = 3 gives us A = 2 and B = 1. The result is:
Linear Factors Raised to a Power
What about a linear factor raised to the second power?
The structure here is:
|We know about linear factors. The new fraction has a single letter over the linear factor. But here we have (x – 1)2.
The power of 2 requires not one, but two new fractions. So, if it were (x – 1)3, we would have three fractions before the (x + 2) fraction (which now has a numerator of D) and they would have as denominators (x – 1), (x – 1)2, and (x -1)3, with corresponding numerators of A, B, and C. The rule for factors raised to the power n is: the right-hand side is the sum of n new fractions where the numerator of each fraction is the typical numerator for this type of factor in PFD, but the denominators are the factor raised to the first power, the second power, and all the way up to the power n.Getting back to our example, the common denominator on the right-hand side is (x – 1)2. Writing each right-hand side fraction over this common denominator and adding gives us: