Algebra (Pure 3)
Partial fractions
- A fraction with a factorised bottom splits into simpler fractions:
$$\frac{1}{(ax+b)(cx+d)} = \frac{A}{ax+b} + \frac{B}{cx+d}$$
- This makes a rational function easy to integrate or expand.
- Example: $\dfrac{x+4}{(x+1)(x-2)} = \dfrac{2}{x-2} - \dfrac{1}{x+1}$.
Practice
Writing (x+4)/((x+1)(x−2)) = A/(x+1) + B/(x−2), substitute x = 2 to find B.
At x = 2: 2 + 4 = B(2 + 1), so 6 = 3B, giving B = 2.
Practice
Partial fractions make a rational function easier to integrate.
Each simple fraction integrates to a logarithm, so splitting first makes integration easy.
Binomial for a rational power
- Works for fractional/negative $n$ when $|x| < 1$:
$$(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \cdots$$
- e.g. $(1+x)^{1/2} = 1 + \tfrac12 x - \tfrac18 x^2 + \cdots$
Practice
The binomial expansion (1+x)ⁿ for a fractional or negative n is valid only when:
For non-positive-integer powers the series converges only for |x| < 1.
You've got it
Key idea
- partial fractions split a rational function for integrating/expanding
- match the form to the bottom (repeated factor → an extra $\dfrac{C}{(cx+d)^2}$ term)
- the binomial works for rational $n$ when $|x| < 1$