Welcome to Partial Fractions!

In your earlier years of maths, you learned how to take two separate fractions and add them together by finding a common denominator. In this chapter, we are going to learn how to do the exact opposite! We will take one large, complicated-looking algebraic fraction and "split" it back into its smaller, simpler parts.

Why is this important? Splitting a fraction makes it much easier to handle in more advanced topics like Integration or Binomial Expansions. Think of it like taking a complex LEGO model apart so you can see exactly which bricks were used to build it.

Note: Based on your MEI H640 syllabus, we will focus on denominators with up to three linear terms, including repeated (squared) terms.


1. Prerequisite Check: Proper vs. Improper Fractions

Before we start splitting, we need to look at the "degree" of the top (numerator) and the bottom (denominator).

  • Degree: The highest power of \(x\) in the expression. For example, \(x^2 + 3\) has a degree of 2.
  • Proper Fraction: The degree of the top is smaller than the degree of the bottom. (e.g., \(\frac{x+1}{x^2+5}\))
  • Improper Fraction: The degree of the top is equal to or larger than the degree of the bottom. (e.g., \(\frac{x^2}{x^2-1}\))

Quick Rule: If a fraction is improper, you must use algebraic long division first to get a whole number/polynomial part plus a proper fraction. Only the proper fraction part gets split into partial fractions!

Key Takeaway: Always check the powers of \(x\) first. If the top power isn't smaller than the bottom power, it's time for long division!


2. Case 1: Distinct Linear Factors

This is the simplest case. It happens when the denominator is made of different linear brackets, like \((x+1)(x-2)\).

The Setup

If you have a fraction like \(\frac{1}{(x-a)(x-b)}\), we assume it came from:
\(\frac{A}{x-a} + \frac{B}{x-b}\)

Step-by-Step Process

  1. Set up the identity: Write your fraction \(\equiv\) the sum of the partial fractions with unknown constants \(A\), \(B\), etc.
  2. Clear the fractions: Multiply every term by the original denominator.
  3. Find the constants:
    • Substitution Method: Pick values of \(x\) that make the brackets equal to zero. This "kills off" all but one constant, letting you solve for it easily.
    • Comparing Coefficients: Match the number of \(x^2\), \(x\), and constants on both sides.

Example: \(\frac{5x-1}{(x-1)(x+2)} \equiv \frac{A}{x-1} + \frac{B}{x+2}\)
Multiply by \((x-1)(x+2)\):
\(5x-1 \equiv A(x+2) + B(x-1)\)
If we let \(x = 1\), the \(B\) term disappears! \(5(1)-1 = A(1+2) \Rightarrow 4 = 3A \Rightarrow A = \frac{4}{3}\).

Quick Review: Distinct factors get their own constant (\(A\), \(B\), \(C\)) over their own bracket.


3. Case 2: Repeated Linear Factors

Don't worry if this seems tricky at first! A repeated factor is just a bracket that is squared, like \((x+1)^2\).

The Setup

If the denominator has \((x-a)^2\), you must include a fraction for every power of that bracket up to the highest one.
\(\frac{P(x)}{(x-a)^2(x-b)} \equiv \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b}\)

Analogy: Imagine climbing a ladder. To get to the second rung \((x-a)^2\), you have to pass the first rung \((x-a)\) as well. You need both in your setup!

Common Mistake: Students often forget the \(\frac{A}{x-a}\) term and only write \(\frac{B}{(x-a)^2}\). Always remember to include the single-power version!

Key Takeaway: For a squared bracket \((x+k)^2\), you need two partial fractions: one with \((x+k)\) and one with \((x+k)^2\).


4. Helpful Tricks and Tips

The "Cover-up" Method

For distinct linear factors, you can find constants quickly. To find \(A\) in \(\frac{5x-1}{(x-1)(x+2)}\):
1. Look at the bracket under \(A\), which is \((x-1)\).
2. What makes it zero? \(x=1\).
3. Cover up \((x-1)\) in the original fraction and plug \(x=1\) into what's left: \(\frac{5(1)-1}{(1+2)} = \frac{4}{3}\).
That's your value for \(A\)! Note: This only works for distinct factors, not repeated ones.

Did you know?

The term "Partial" fractions simply means we are breaking a "Whole" fraction into "Parts". It’s like breaking the number 10 into 7 + 3.


Summary Checklist

  • Is it proper? If the top power \(\ge\) bottom power, use long division first.
  • Factorise the bottom: Make sure the denominator is fully factorised into linear brackets.
  • Choose the right setup:
    • Distinct: \(\frac{A}{x-a} + \frac{B}{x-b}\)
    • Repeated: \(\frac{A}{x-a} + \frac{B}{(x-a)^2}\)
  • Substitute "Smart" values: Use values of \(x\) that make your brackets zero to solve for your constants quickly.
  • Double check: Pick a random value for \(x\) (like \(x=0\)) and plug it into both the original and your partial fractions to see if they match!