IP AMaths Notes (Upper Sec, Year 3-4): 18) Partial Fractions

Partial fractions simplify rational functions for integration and algebraic manipulation.

Templates

• Distinct linear factors: \( \dfrac{P(x)}{(x - a)(x - b)} = \dfrac{A}{x - a} + \dfrac{B}{x - b} \).
• Repeated: \( \dfrac{P(x)}{(x - a)^2} = \dfrac{A}{x - a} + \dfrac{B}{(x - a)^2} \).
• Irreducible quadratic: \( \dfrac{P(x)}{(x^{2} + px + q)} = \dfrac{Ax + B}{x^{2} + px + q} \).

Worked example 1 — Distinct factors

Decompose \( \dfrac{7x + 5}{x^{2} + 3x - 4} \).

1. Factor denominator: \( x^{2} + 3x - 4 = (x + 4)(x - 1) \).
2. Set \( \dfrac{7x + 5}{(x + 4)(x - 1)} = \dfrac{A}{x + 4} + \dfrac{B}{x - 1} \).
3. Multiply both sides by denominator: \( 7x + 5 = A(x - 1) + B(x + 4) \).
4. Substitute \( x = 1 \): \( 12 = 5B \) → \( B = \tfrac{12}{5} \).
5. Substitute \( x = -4 \): \( -23 = -5A \) → \( A = \tfrac{23}{5} \).
6. Therefore \( \dfrac{7x + 5}{x^{2} + 3x - 4} = \dfrac{23}{5(x + 4)} + \dfrac{12}{5(x - 1)} \).

Worked example 2 — Repeated factor

Decompose \( \dfrac{2x + 1}{x^{2}(x + 3)} \).

1. Form: \( \dfrac{2x + 1}{x^{2}(x + 3)} = \dfrac{A}{x} + \dfrac{B}{x^{2}} + \dfrac{C}{x + 3} \).
2. Multiply through: \( 2x + 1 = A x(x + 3) + B(x + 3) + C x^{2} \).
3. Substitute \( x = 0 \): \( 1 = 3B \) → \( B = \tfrac{1}{3} \).
4. Substitute \( x = -3 \): \( -5 = C \times 9 \) → \( C = -\tfrac{5}{9} \).
5. Compare coefficients for \( x^{2} \): left side has zero; right side gives \( A + C = 0 \) so \( A = -C = \tfrac{5}{9} \).
6. Final form: \( \dfrac{2x + 1}{x^{2}(x + 3)} = \dfrac{5}{9x} + \dfrac{1}{3x^{2}} - \dfrac{5}{9(x + 3)} \).

Try this

Decompose \( \dfrac{5x^{2} + 7x - 3}{(x - 1)^2(x + 2)} \) and note the coefficients clearly.