Algebra (Pure 3)

Splitting the unsplitable

• You can't integrate $\dfrac{1}{x^2 - 1}$ directly. But split it into $\dfrac{1}{2(x-1)} - \dfrac{1}{2(x+1)}$ and each piece is a simple logarithm.
• Partial fractions 部分分式 turn hard rational functions 有理函数 into easy pieces — essential for integration and series expansion.

Partial fractions

• A fraction with a factorised 因式分解 bottom splits into simpler fractions:

• This makes a rational function easy to integrate or expand.

The scalar product measures how much of b points along a: |b| cos theta

Repeated and quadratic factors

• Repeated factor 重复因式 $(cx+d)^2$: needs $\dfrac{B}{cx+d} + \dfrac{C}{(cx+d)^2}$.
• Irreducible quadratic 不可约二次式 $(x^2 + a^2)$: needs $\dfrac{Bx + C}{x^2 + a^2}$.

Holiday figure

Binomial expansion 二项展开式 for rational powers

• Works for fractional/negative $n$ when $|x| < 1$:

• e.g. $(1+x)^{1/2} = 1 + \tfrac12 x - \tfrac18 x^2 + \cdots$

Worked example — integrating with partial fractions

• $\displaystyle\int \frac{x+4}{(x+1)(x-2)}\,dx = \int\left(\frac{2}{x-2} - \frac{1}{x+1}\right)dx = 2\ln|x-2| - \ln|x+1| + C$.

allow_no_figure: Algebraic technique (partial fractions decomposition) — the visual content is the algebraic manipulation itself, not a diagram.

• Use the factor theorem and remainder theorem: dividing a polynomial gives a quotient and a remainder.

You've got it

Write from the numerators 3 and -3

• Split 3 divided by x(x + 1) into partial fractions.