Key Idea: Partial fractions split one fraction into two simpler ones. It is by-hand algebra, because a GDC will not split it for you. It is on Paper 1 and Paper 2.
Splitting into two fractions
- Write the split — Two different linear factors below: A/(x − a) + B/(x − b).
- Clear fractions — Multiply both sides by the whole bottom.
- Cover up — Put in each bracket's root. Each one gives you A, then B.
- Check — Add your two fractions back. You should get the original.
Getting it ready first
- Bottom not in brackets? Factorise it, e.g. x² − 4 = (x − 2)(x + 2).
- Top degree as big as the bottom degree? Divide first, then split what is left.
- Instead of cover-up you can equate coefficients. You get the same A and B.
Only a proper fraction with a factorised bottom can be split.
Why does putting x = 2 into 4x − 5 = A(x − 3) + B(x − 2) give only A? The B bracket becomes 0, so B drops out.
What do you do first with (5x − 4)/(x² − 4)? Factorise the bottom: (x − 2)(x + 2).
The top has a bigger degree than the bottom. What now? Divide first to take out a whole part.
How can you check your answer? Add the two fractions back together.