Partial fractions — A/L Combined Mathematics

What must I remember about Partial fractions?

• A rational function is f(x) = P(x) / Q(x) where P(x) and Q(x) are polynomials with Q(x) ≠ 0; it is Proper if deg(P) < deg(Q), and Improper if deg(P) ≥ deg(Q).
• Improper rational functions must first be divided using polynomial long division: P(x)/Q(x) = S(x) + R(x)/Q(x) where S(x) is a polynomial quotient and deg(R) < deg(Q).
• Linear non-repeated factors in denominator: If Q(x) contains factor (ax + b), the corresponding partial fraction term is A / (ax + b).
• Repeated linear factors: If Q(x) contains (ax + b)^k, the partial fractions include A1/(ax + b) + A2/(ax + b)² + ... + Ak/(ax + b)^k.
• Irreducible quadratic factors: If Q(x) contains (ax² + bx + c) where b² - 4ac < 0, the corresponding partial fraction term is (Ax + B) / (ax² + bx + c).

Test yourself (1 minute)

1. Cover-up rule (Heaviside method)

   For distinct linear factors (x - a), the numerator coefficient A is obtained instantly by covering up (x - a) in the original fraction and substituting x = a into the remaining expression.

2. Degrees check before decomposition

   Always verify that deg(P) < deg(Q); attempting partial fractions directly on an improper fraction produces erroneous results lacking the polynomial term.

3. Coefficient comparison vs substitution

   Use substitution of root values x = a, b to find constants quickly, and compare highest power (or constant) coefficients to solve for remaining parameters.

Practise Partial fractions questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 5

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Where does Partial fractions fit in the syllabus?

Year 1 · Term 1