Use the general term to find a coefficient
Select the required term from a binomial expansion using the general term, without expanding every term first.
Section 12 · Series
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which values can r take in the general term of (a + b)ⁿ?
What you will learn
- Use the general term C(n,r)aⁿ⁻ʳbʳ with 0 ≤ r ≤ n
- Find the coefficient of a specified power of x
- Simplify the binomial coefficient and variable powers accurately
Match the variable exponent to the requested term
For (a + b)ⁿ, the general term is C(n,r)aⁿ⁻ʳbʳ, where r is an integer from 0 to n. If a or b contains x, combine its exponents to find the power of x in that term.
Write the x-exponent as an expression involving r, set it equal to the requested exponent, solve for an allowed integer r, then calculate only that coefficient.
The general term can also identify an x-independent term
In (x² + 1/x)⁶, the general term is C(6,r)(x²)⁶⁻ʳ(x⁻¹)ʳ = C(6,r)x¹²⁻³ʳ. The power is zero when r = 4, so the independent term is C(6,4) = 15.
Check that the value of r is in the permitted range 0 ≤ r ≤ n. If no integer r satisfies the target exponent, that power does not appear.
Pause and explain
If the x-power in a term is 7 − r and the target is x³, what should you solve?
Worked example
Find the coefficient of x⁴ in (2x − 3)⁶.
Show the worked solution
- The general term is C(6,r)(2x)⁶⁻ʳ(−3)ʳ, so the power of x is 6 − r.
- Set 6 − r = 4 to get r = 2, which is allowed.
- The coefficient is C(6,2) × 2⁴ × (−3)² = 15 × 16 × 9.
Answer The coefficient of x⁴ is 2160.
Try a topic test
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Test this topic →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.
Find the coefficient of x² in (2x + 1)⁵.
Give me a hint
The general term has x-power 5 − r.
Compare my reasoning
- Set 5 − r = 2, so r = 3.
- The coefficient is C(5,3) × 2² × 1³.
- Calculate 10 × 4.
The coefficient of x² is 40.
Look for these in your work
- Correct term index selected
- Binomial coefficient used
- Numerical coefficient simplified
Find the term independent of x in (x² + 1/x)⁶.
Give me a hint
Write the x-exponent in the general term and set it to zero.
Compare my reasoning
- The term is C(6,r)x¹²⁻³ʳ.
- Set 12 − 3r = 0 to get r = 4.
- The coefficient is C(6,4) = 15.
The independent term is 15.
Look for these in your work
- General term formed
- Exponent set to zero
- Allowed integer index used
Find the coefficient of x⁰ in (x³ + 2/x)⁵.
Give me a hint
The general term's x-power is 3(5 − r) − r.
Compare my reasoning
- Simplify the exponent to 15 − 4r.
- Set 15 − 4r = 0; this gives r = 15/4, which is not an integer.
- No term has that index, so the expansion contains no x-independent term.
The coefficient of x⁰ is 0 because no integer r from 0 to 5 gives exponent zero.
Look for these in your work
- Both variable powers included
- Index condition solved
- Non-integer index interpreted correctly
Common mistakes
- Using r as the term number instead of the number of b factors
- Ignoring the power of x inside either binomial term
- Solving for a value of r outside 0 ≤ r ≤ n
- Including the variable when the question asks for its coefficient
What can you explain now?
In (x + 2)⁵, what is the power of x in the term indexed by r?
Compare with the explanation
5 − r
The factor x is raised to n − r = 5 − r.
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Make it stick
How do you use an exponent equation to select one term from a binomial expansion?
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.