A Level · Cambridge (CIE) · 9709

Decompose distinct linear factors and equal-degree rational functions

Choose one constant numerator per distinct linear factor, find the coefficients and rebuild the original fraction.

Pure Mathematics 3 · Algebra · 3.1

Exam scope and assessment notes

Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027, section 3.1. Paper 1 knowledge is assumed. Shared modulus, polynomial division, factor and remainder lessons are linked from both sections 2.1 and 3.1. Paper 3 adds partial fractions and rational-power binomial expansions. Partial-fraction numerators may have the same degree as the denominator, but higher-degree numerators are excluded. General-term questions for the rational binomial expansion are not included. Use the stated strict validity condition. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Factor polynomials and use a quotient–remainder identity.
Solve simultaneous linear equations and evaluate polynomials at signed or fractional inputs.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

At which input is 2x − 1 zero?

What you will learn

Decompose proper fractions with up to three distinct linear factors, including non-monic factors.
Extract a constant quotient when the numerator and denominator have equal degree; retain all original domain exclusions.

Choose a numerator of lower degree than its denominator factor. Three distinct linear factors need constant numerators A, B and C. A repeated linear factor needs terms for both its first and second powers. An irreducible quadratic Q needs the linear numerator Bx + C. Here L, L₁, L₂ and L₃ denote linear factors, and Q denotes an irreducible quadratic. These are templates for proper fractions; first extract a constant quotient if the whole numerator and denominator have equal degree. Keep the original denominator exclusions when recombining.

Choose the form before finding its coefficients

For a denominator containing three distinct linear factors L₁, L₂ and L₃, use A/L₁ + B/L₂ + C/L₃, with constants A, B and C. A two-factor case uses two terms. A proper fraction has a numerator of lower degree than its denominator. Multiply by the full denominator to obtain a polynomial identity; this identity holds at all real inputs even though the original fractions exclude the denominator's zeros.

For (5x + 1)/[(x − 1)(x + 2)], the identity is 5x + 1 = A(x + 2) + B(x − 1). At x = 1, 6 = 3A, so A = 2. At x = −2, −9 = −3B, so B = 3. Hence the fraction is 2/(x − 1) + 3/(x + 2), for x ≠ 1, −2. Recombining the numerators gives 2(x + 2) + 3(x − 1) = 5x + 1.

A non-monic factor changes its root, not the partial-fraction rule

For (5x − 1)/[(2x − 1)(x + 1)], start with A/(2x − 1) + B/(x + 1). The identity is 5x − 1 = A(x + 1) + B(2x − 1). Use x = 1/2 to get A = 1, and x = −1 to get B = 2. Substituting 1 instead of 1/2 into the first root calculation would not eliminate the other coefficient. Always solve the linear factor's actual equation.

If the numerator and denominator have equal degree, first divide to remove a constant quotient, then decompose the proper remainder fraction. For (x² + 4x − 1)/[(x − 1)(x + 2)], the denominator is x² + x − 2, so division gives 1 + (3x + 1)/[(x − 1)(x + 2)]. The remainder decomposes as (4/3)/(x − 1) + (5/3)/(x + 2). A sum of proper partial fractions alone cannot reproduce the leading constant 1. Higher-degree numerators are outside this section's specified partial-fraction cases.

Pause and explain

What must precede proper partial fractions for (x² + 4x − 1)/(x² + x − 2)?

Put the idea to work

Worked example

Decompose (5x + 1)/[(x − 1)(x + 2)] and state the exclusions.

Show the worked solution

Write A/(x − 1) + B/(x + 2) and clear the denominator.
The identity 5x + 1 = A(x + 2) + B(x − 1) gives A = 2 at 1 and B = 3 at −2.
Recombine to confirm 5x + 1 and retain x ≠ 1, −2.

Answer 2/(x − 1) + 3/(x + 2), with x ≠ 1, −2.

Try a topic test

Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

Decompose (5x + 1)/[(x − 1)(x + 2)] and verify its numerator.

Give me a hint

Write both constant-numerator terms and substitute each factor's zero into the cleared identity.

Compare my reasoning

Write 5x + 1 = A(x + 2) + B(x − 1).
At 1, A = 2; at −2, B = 3.
The rebuilt numerator is 2(x + 2) + 3(x − 1) = 5x + 1; exclude both denominator roots.

2/(x − 1) + 3/(x + 2), x ≠ 1, −2.

Look for these in your work

Find both constants from the identity.
Recombine the numerator and retain the original domain.

2 · Independent

Decompose (2x² + 6x − 2)/[x(x − 1)(x + 2)].

Give me a hint

Use a term for each of the three distinct linear factors and evaluate at their three roots.

Compare my reasoning

Set the numerator equal to A(x − 1)(x + 2) + Bx(x + 2) + Cx(x − 1).
At the three factor roots 0, 1 and −2, obtain A = 1, B = 2 and C = −1.
Expanding (x − 1)(x + 2) + 2x(x + 2) − x(x − 1) gives 2x² + 6x − 2; exclude 0, 1 and −2.

1/x + 2/(x − 1) − 1/(x + 2), x ≠ 0, 1, −2.

Look for these in your work

Include all three denominator factors.
Verify the complete polynomial numerator, including its constant term.

3 · Transfer

Express (x² + 4x − 1)/[(x − 1)(x + 2)] as a constant plus partial fractions, and evaluate it at x = 0.

Give me a hint

Divide the equal-degree polynomials first, then decompose only the remainder fraction.

Compare my reasoning

Division gives constant 1 and remainder numerator 3x + 1.
From 3x + 1 = A(x + 2) + B(x − 1), the constants are A = 4/3 and B = 5/3.
At 0, 1 − 4/3 + 5/6 = 1/2, matching (−1)/(−2). The excluded inputs remain 1 and −2.

1 + 4/[3(x − 1)] + 5/[3(x + 2)]; value 1/2 at 0; x ≠ 1, −2.

Look for these in your work

Include the constant quotient before decomposing.
Check the original fraction at an allowed input.

Common mistakes

Use one term for each distinct factor, even if one coefficient later turns out to be zero.
Find the actual root of a non-monic factor.
An equal-degree fraction needs a constant quotient before its proper part is decomposed.

Recall without your notes

What can you explain now?

Decompose (5x − 1)/[(2x − 1)(x + 1)].

Compare with the explanation

1/(2x − 1) + 2/(x + 1), x ≠ 1/2, −1.

Clearing the denominator gives (x + 1) + 2(2x − 1) = 5x − 1; factor roots are 1/2 and −1.

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Make it stick

Explain why each distinct factor needs a constant numerator, how a cleared identity can be evaluated at excluded inputs, and why equal-degree division comes first.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.