A Level · Cambridge (CIE) · 9709

Include every power of a repeated linear factor in partial fractions

Use both simple and squared denominator terms, then obtain the remaining coefficient by comparing the identity.

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

Decompose fractions with distinct linear factors.
Expand quadratic products and compare corresponding coefficients.

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

Quick readiness check

How many terms are needed for the general denominator (x − 1)(x + 1)²?

What you will learn

Use A/L₁ + B/L₂ + C/L₂² when one linear factor is repeated.
Find all coefficients, including those not isolated by root substitution, and verify the resulting identity.

Repeated factors need a term for each power

For a denominator (x − 1)(x + 1)², the complete form is A/(x − 1) + B/(x + 1) + C/(x + 1)². Omitting the simple (x + 1) term usually makes the identity impossible. After clearing the denominator, the numerator is A(x + 1)² + B(x − 1)(x + 1) + C(x − 1). The squared-denominator term contributes only one remaining linear factor to this identity.

For numerator 2x² − x + 3, substitute x = 1 to get 4 = 4A, so A = 1. Substitute x = −1 to get 6 = −2C, so C = −3. Neither substitution finds B: both make its product zero. Comparing x² coefficients gives 2 = A + B, hence B = 1. The decomposition is 1/(x − 1) + 1/(x + 1) − 3/(x + 1)², excluding 1 and −1.

Finish and verify the whole polynomial identity

Expand the result: (x + 1)² + (x − 1)(x + 1) − 3(x − 1) = 2x² − x + 3. This checks all coefficients, not only the two substituted values. A different convenient input, such as 0, also supplies the final equation if coefficient comparison is inconvenient. Keep signs attached to the full products when expanding.

For a denominator (x − 1)² alone, use A/(x − 1) + B/(x − 1)². Thus (3x − 1)/(x − 1)² has identity 3x − 1 = A(x − 1) + B, giving A = 3 and B = 2. A non-monic repeated factor follows the same template; its zero is found by solving its linear equation. Repeated denominator factors do not restore any excluded input, even if some coefficient is zero.

Pause and explain

In the cleared identity, root substitution gives A = 1 and C = −3. How can B be found?

Put the idea to work

Worked example

Decompose (2x² − x + 3)/[(x − 1)(x + 1)²].

Show the worked solution

Use A/(x − 1) + B/(x + 1) + C/(x + 1)² and clear the denominator.
At x = 1 obtain A = 1; at x = −1 obtain C = −3.
Compare x² coefficients to get B = 1, then expand to verify the numerator.

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

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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 (3x − 1)/(x − 1)².

Give me a hint

Include both denominator powers and compare the linear coefficient and constant term.

Compare my reasoning

Write A/(x − 1) + B/(x − 1)².
Clear the denominator: 3x − 1 = A(x − 1) + B; the x coefficient gives A = 3.
The constant equation −1 = −3 + B gives B = 2. Recombination recovers 3x − 1.

3/(x − 1) + 2/(x − 1)², x ≠ 1.

Look for these in your work

Include both first and second powers.
Match both coefficients in the cleared identity.

2 · Independent

Decompose (2x² − x + 3)/[(x − 1)(x + 1)²], explaining how the middle coefficient is determined.

Give me a hint

After root substitution, use the leading coefficient or another allowed input for the remaining constant.

Compare my reasoning

The complete form has coefficients A, B and C over x − 1, x + 1 and (x + 1)².
At 1 obtain A = 1; at −1 obtain C = −3. Compare x² coefficients: 2 = A + B, so B = 1.
Rebuild the numerator as (x + 1)² + (x − 1)(x + 1) − 3(x − 1) = 2x² − x + 3.

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

Look for these in your work

Find the coefficient not isolated by root evaluation.
Verify all polynomial coefficients and domain exclusions.

3 · Transfer

Decompose (5x² − 2x + 2)/[(2x − 1)(x + 1)²].

Give me a hint

Use the actual non-monic root 1/2, then compare leading coefficients to finish.

Compare my reasoning

Clear the denominator using A(x + 1)² + B(2x − 1)(x + 1) + C(2x − 1).
At 1/2 obtain A = 1; at −1 obtain C = −3. The x² equation is 5 = A + 2B, so B = 2.
Expand (x + 1)² + 2(2x − 1)(x + 1) − 3(2x − 1) to recover 5x² − 2x + 2.

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

Look for these in your work

Handle the non-monic linear factor without a scale error.
Use the leading coefficient to complete and verify the form.

Common mistakes

Do not skip a lower power of a repeated factor.
Root substitution may leave a coefficient undetermined; add an independent equation.
Check the entire polynomial identity after finding all coefficients.

Recall without your notes

What can you explain now?

What is the full form for a proper fraction with denominator (2x − 1)(x + 1)²?

Compare with the explanation

A/(2x − 1) + B/(x + 1) + C/(x + 1)².

The distinct linear factor needs one term and the repeated linear factor needs both its first and second powers.

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Make it stick

Write the complete repeated-factor template from memory, then explain why two factor-root substitutions can leave the middle coefficient unknown.

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.