Partial fractions express a rational function $\dfrac{P(x)}{Q(x)}$ (with $\deg P < \deg Q$) as a sum of simpler fractions. This is essential for integration of rational functions.
Every polynomial over $\mathbb{R}$ factors into linear and irreducible quadratic factors:
$$Q(x) = a(x-r_1)^{m_1}\cdots(x-r_k)^{m_k}(x^2+p_1 x+q_1)^{n_1}\cdots$$
If $\deg P \geq \deg Q$, perform polynomial division first.
$$\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$$
Multiply through and compare coefficients (or substitute convenient $x$ values).
Example: $\dfrac{3x+1}{(x-1)(x+2)} = \dfrac{A}{x-1} + \dfrac{B}{x+2}$.
Multiply: $3x+1 = A(x+2) + B(x-1)$.
$x=1$: \$4 = 3A \Rightarrow A = 4/3$.
$x=-2$: $-5 = -3B \Rightarrow B = 5/3$.
$$\frac{3x+1}{(x-1)(x+2)} = \frac{4/3}{x-1} + \frac{5/3}{x+2}$$
$$\frac{P(x)}{(x-a)^n} = \frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n}$$
Example: $\dfrac{x+3}{(x-1)^2(x+1)} = \dfrac{A}{x-1} + \dfrac{B}{(x-1)^2} + \dfrac{C}{x+1}$.
Multiply: $x+3 = A(x-1)(x+1) + B(x+1) + C(x-1)^2$.
$x=1$: \$4 = 2B \Rightarrow B = 2$.
$x=-1$: \$2 = 4C \Rightarrow C = 1/2$.
Coefficient of $x^2$: \$0 = A + C \Rightarrow A = -1/2$.
$$\frac{x+3}{(x-1)^2(x+1)} = \frac{-1/2}{x-1} + \frac{2}{(x-1)^2} + \frac{1/2}{x+1}$$
An irreducible quadratic $x^2 + px + q$ has $\Delta < 0$. Assign a linear numerator:
$$\frac{P(x)}{(x-a)(x^2+bx+c)} = \frac{A}{x-a} + \frac{Bx+C}{x^2+bx+c}$$
Example: $\dfrac{2x^2+x+1}{(x-1)(x^2+x+1)} = \dfrac{A}{x-1} + \dfrac{Bx+C}{x^2+x+1}$.
($x^2+x+1$ has $\Delta = 1-4 = -3 < 0$, so it is irreducible.)
Multiply: $2x^2+x+1 = A(x^2+x+1) + (Bx+C)(x-1)$.
$x=1$: \$4 = 3A \Rightarrow A = 4/3$.
Expand RHS: $(A+B)x^2 + (A-B+C)x + (A-C)$.
Compare $x^2$: \$2 = A+B \Rightarrow B = 2/3$.
Compare constant: \$1 = A-C \Rightarrow C = A-1 = 1/3$.
$$\frac{P(x)}{(x^2+a)^2} = \frac{Ax+B}{x^2+a} + \frac{Cx+D}{(x^2+a)^2}$$
Partial fractions enable integration:
$$\int \frac{A}{x-a}\,dx = A\ln|x-a| + c$$
$$\int \frac{Bx+C}{x^2+px+q}\,dx \quad \text{(complete the square, then arctan or log forms)}$$
Quick integration example:
$$\int \frac{3x+1}{(x-1)(x+2)}\,dx = \frac{4}{3}\ln|x-1| + \frac{5}{3}\ln|x+2| + c$$
KEY TAKEAWAY: The form of partial fractions depends on the type and multiplicity of factors. Linear factors give constant numerators; irreducible quadratic factors give linear numerators; repeated factors give multiple terms.
EXAM TIP: Use the substitution method (setting $x$ equal to roots) to find constants quickly. For remaining constants, compare coefficients of a specific power of $x$.
COMMON MISTAKE: Assigning a constant numerator to an irreducible quadratic factor. The numerator must be linear: $Bx + C$.
VCAA FOCUS: Partial fractions appear in integration questions. Confirm the denominator is fully factorised and check $\deg P < \deg Q$ before setting up the decomposition.
