Systems of equations and partial fractions (IB Math AA HL)

Key ideas

• Partial fractions need the degree of the numerator to be less than the degree of the denominator; divide first if it is not.
• For distinct linear factors: (px + q)/((x − a)(x − b)) = A/(x − a) + B/(x − b).
• Cover-up: to find A, cover (x − a) and substitute x = a into what is left. Check by recombining the fractions.
• Each part integrates to a log: ∫A/(x − a) dx = A ln|x − a| + C. Combine the logs at the end if an exact answer is asked for.
• A 3 × 3 linear system can have one solution, none or infinitely many; with a parameter, find when the determinant is zero, then check for consistency.

Worked example

Express (5x + 1)/((x − 1)(x + 2)) in partial fractions, and hence find the exact value of ∫₂⁴ (5x + 1)/((x − 1)(x + 2)) dx.

1. Write (5x + 1)/((x − 1)(x + 2)) = A/(x − 1) + B/(x + 2), so 5x + 1 = A(x + 2) + B(x − 1).
2. x = 1: 6 = 3A, so A = 2. x = −2: −9 = −3B, so B = 3.
3. Integral = [2 ln|x − 1| + 3 ln|x + 2|] from 2 to 4.
4. = (2 ln 3 + 3 ln 6) − (2 ln 1 + 3 ln 4) = 2 ln 3 + 3 ln(3/2).
5. = ln 9 + ln(27/8) = ln(243/8) ≈ 3.41.

Answer: 2/(x − 1) + 3/(x + 2); the integral is ln(243/8) ≈ 3.41.

Exam-style HL AA questions on this topic

• Partial Fractions questions — 20 questions with mark schemes, plus free worked examples
• Systems of Equations questions — 30 questions with mark schemes, plus free worked examples
• All HL AA Number and Algebra questions by topic

Do it on your GDC

A calculator is allowed on AA Paper 2 and Paper 3, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Solving equations on your calculator

Related in Number and Algebra

• Proof by induction — the 3-step method
• Binomial theorem for rational and negative exponents
• Complex numbers — Cartesian, polar, and Euler forms

Revise Systems of equations and partial fractions for the exam

• HL AA revision notes — every unit summarised, with the key results to learn
• HL AA past papers — specimen papers and timed practice papers
• Formulas for Number and Algebra — on the one-page HL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S93 · Problem S28 · Problem S92 · Problem S27. All 22 →

Sibling topic in other IB Maths courses