A-Level — mathsAdvanced1 min read

Partial Fractions

Decompose algebraic fractions into partial fractions at A-Level. Handle linear, repeated, and irreducible factors.

Tutor AI Team2026-02-25T01:53:03.156Z

Partial fractions decompose a complex fraction into simpler ones. This is essential for integration and series work at A-Level.

Types

$\frac{f(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$

$\frac{f(x)}{(x-a)^2(x-b)} = \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b}$

If degree of numerator ≥ degree of denominator, divide first.

$\frac{5x + 1}{(x+1)(x-2)}$

Let $x = -1$ : $\frac{-4}{-3} = \frac{4}{3} = A$ .

Let $x = 2$ : $\frac{11}{3} = B$ .

So $\frac{5x+1}{(x+1)(x-2)} = \frac{4/3}{x+1} + \frac{11/3}{x-2}$ .

Equate numerators and compare powers of $x$ .

$\frac{3x + 5}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}$

$3x + 5 = A(x+2) + B(x+1)$

$x = -1$ : $2 = A(1)$ → $A = 2$ . $x = -2$ : $-1 = B(-1)$ → $B = 1$ .

Answer: $\frac{2}{x+1} + \frac{1}{x+2}$ .

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#a-level#maths#pure#algebra#partial-fractions

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