Expand a positive-integer binomial
Use the binomial theorem to expand (a + b)ⁿ for a positive integer n and simplify each coefficient and power.
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
What is the general term in (a + b)ⁿ?
What you will learn
- Apply the binomial theorem for a positive integer power
- Use binomial coefficients to generate each term
- Simplify signs, numerical coefficients and powers of variables
Each term chooses how many factors contribute b
The expansion of (a + b)ⁿ is the sum of C(n,r)aⁿ⁻ʳbʳ for r = 0, 1, …, n. The first term has r = 0 and the final term has r = n; there are n + 1 terms.
The coefficient row begins 1, n, …, n, 1 and is symmetric. The formula for the binomial theorem is supplied in the 0606 examination, but each power and sign must still be simplified correctly.
Track the exponents and signs together
For (2x + 3)⁴, the rth term is C(4,r)(2x)⁴⁻ʳ3ʳ. As r increases, the power of x falls by one and the power of 3 rises by one.
For a negative term such as (3x − 1)³, use b = −1 in every term. The sign alternates because (−1)ʳ changes sign when r changes parity.
Pause and explain
What is the first term of (x + 2)⁴?
Worked example
Expand (2x + 3)⁴ and simplify.
Show the worked solution
- Use coefficients 1, 4, 6, 4, 1 and powers (2x)⁴⁻ʳ3ʳ.
- The five terms are 16x⁴, 96x³, 216x², 216x and 81.
- Arrange them in descending powers of x.
Answer 16x⁴ + 96x³ + 216x² + 216x + 81.
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.
Expand (x + 2)⁴.
Give me a hint
Use coefficients 1, 4, 6, 4, 1 and let the power of 2 increase.
Compare my reasoning
- Write x⁴ + 4x³(2) + 6x²(2²) + 4x(2³) + 2⁴.
- Evaluate the numerical factors: 8, 24, 32 and 16.
- Keep the terms in descending powers of x.
x⁴ + 8x³ + 24x² + 32x + 16.
Look for these in your work
- Five terms included
- Coefficients and powers matched
- Descending powers retained
Expand (3x − 1)³.
Give me a hint
Use coefficients 1, 3, 3, 1 and keep (−1)ʳ in each term.
Compare my reasoning
- Write (3x)³ + 3(3x)²(−1) + 3(3x)(−1)² + (−1)³.
- Evaluate each term: 27x³, −27x², 9x and −1.
- Check the alternating signs and descending powers.
27x³ − 27x² + 9x − 1.
Look for these in your work
- Four terms included
- Alternating signs correct
- Powers and coefficients simplified
Expand (2 − x)⁵ and state the coefficient of x³.
Give me a hint
Treat the second term as −x; its power controls the sign.
Compare my reasoning
- Use coefficients 1, 5, 10, 10, 5, 1.
- The expansion is 32 − 80x + 80x² − 40x³ + 10x⁴ − x⁵.
- Read the x³ coefficient, including its sign.
32 − 80x + 80x² − 40x³ + 10x⁴ − x⁵; coefficient of x³ is −40.
Look for these in your work
- Six terms included
- Alternating signs checked
- Requested coefficient identified
Common mistakes
- Stopping after n terms instead of n + 1
- Raising the coefficient but forgetting to raise x
- Losing the alternating signs when b is negative
- Writing the terms in a different power order without checking them
What can you explain now?
How many terms appear in the expansion of (a + b)⁷?
Compare with the explanation
8
The powers use r = 0 through r = 7, giving eight terms.
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Make it stick
How do the coefficient and exponents change as r runs from 0 to n in the 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.