Algebra: partial fractions, series, binomial expansion

Pure Mathematics 3 – Algebra: Partial Fractions, Series, Binomial Expansion

Partial Fractions 🍕

Think of a rational function as a pizza that you want to slice into simpler pieces. If you can break the pizza into slices that are easier to eat (simplify), you can also integrate or differentiate each slice separately.

1. Factor the denominator into linear and/or irreducible quadratic factors.
2. Write the fraction as a sum of unknown constants over each factor.
3. Clear the denominator by multiplying through.
4. Match coefficients or plug in convenient values to solve for the unknowns.

Example: $$\frac{2x+3}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$$ Multiply both sides by $(x-1)(x+2)$ and solve for $A$ and $B$.

Series 📈

A series is like a long line of dominoes. If you know the pattern of each domino, you can predict how many will fall in total (the sum).

• Geometric series: each term is a constant multiple of the previous one.
• Harmonic series: terms are reciprocals of natural numbers.
• Power series: terms involve powers of a variable.

Geometric series sum: $$\displaystyle \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}\quad (|r|<1)$$

Binomial Expansion 🔢

The binomial theorem lets you expand powers of a binomial (like $(x+y)^n$) into a sum of terms. Think of it as a recipe: each term is a flavour that appears a specific number of times.

Binomial theorem: $$\displaystyle (x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{\,n-k} y^{\,k}$$

The coefficients $\binom{n}{k}$ are read from Pascal’s Triangle.

Example: $(x+y)^4 = 1x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + 1y^4$.