Top-band candidates ALWAYS state the range of validity.
3Partial fractions β repeated factor (8 marks)
Extendedβ’ partial fractions
βΌ
Question
Express (xβ1)22x+1β in partial fractions. (8 marks)
Step-by-step solution
Step 1
Form for repeated factor.xβ1Aβ+(xβ1)2Bβ.
Step 2
Multiply by (xβ1)2.
2x+1=A(xβ1)+B
Step 3
Substitute x=1.3=0+BβB=3.
Step 4
Compare x coefficients.2=A.
Step 5
Final.
(xβ1)22x+1β=xβ12β+(xβ1)23β
Answer
xβ12β+(xβ1)23β.
Key Formulae β Algebra
The formulae you need to memorise for algebra on the Cambridge International A Level 9709 paper, with every variable defined in plain English and a note on when to use it.
Decomposing rational function. Three forms cover distinct linear, repeated linear, and quadratic factors.
Key Definitions and Keywords β Algebra
Definitions to memorise and the exact keywords mark schemes credit for algebra answers β sharpened from recent examiner reports for the 2026 Cambridge International A Level 9709 sitting.
Partial fractions
Examiner keyword
Decomposing a single rational function into sum of simpler fractions with denominators that are factors of the original.
Binomial expansion (general n)
Examiner keyword
Infinite series expansion of (1+x)n for any real n, valid when β£xβ£<1.
Range of validity
Examiner keyword
Values of x for which a binomial series converges. For (1+ax)n: β£axβ£<1.
Common Mistakes and Misconceptions β Algebra
The traps other students keep falling into on algebra questions β taken from recent Cambridge International A Level 9709 examiner reports and mark schemes β and how to avoid them.
βWrong partial-fraction form for repeated factor
9709 Examiner Reports 2022-2024
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Why it happens
Confusing with distinct linear case.
How to avoid it
Repeated factor (xβa)2: form is xβaAβ+(xβa)2Bβ. Need BOTH terms.
βForgetting validity condition for binomial expansion
9709 Examiner Reports 2022-2024
βΌ
Why it happens
Habit from P1 binomial (positive integer, always valid).
How to avoid it
P3 binomial (any real n) requires β£axβ£<1. State this explicitly.
Practice questions
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Video lesson
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Algebra β frequently asked questions
The things students keep getting wrong in this sub-topic, answered.