Select binomial coefficients and solve for unknown parameters
Choose the correct power term, include contributions from products and retain every parameter value consistent with the given coefficients.
Pure Mathematics 1 (Paper 1) · AS & A Level Mathematics 9709, 2026–2027. Use positive-integer binomial powers and positive-integer term indices. Written practice is self-checked.
Before you begin
- Use the general term of a positive-integer binomial expansion.
- Solve quadratic equations and collect like powers in a product.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which term of (a + bx)^n supplies x² when b is constant?
What you will learn
- Find a requested coefficient without expanding unnecessary terms.
- Use supplied coefficient information to solve for a parameter and check possible signs.
Match the exponent before calculating the coefficient
For (2 + x)^5, the x³ term has k = 3 and coefficient (5 choose 3)2² = 10 × 4 = 40. For (3 − 2x)^5, the x² coefficient is (5 choose 2)3³(−2)² = 1080. The term's position is k + 1, so an x² term is usually the third term when the variable occurs as x, rather than the second.
When the variable term is x², its kth power is x^(2k). Thus (1 + x²)^4 has no x³ term. In a product, more than one pair of powers can contribute: the x² coefficient of (1 + x)(2 − x)^4 combines 1 times the x² coefficient 24 and x times the x coefficient −32, giving 24 − 32 = −8. Count every route to the target power.
A squared parameter can have two signs
In (1 + kx)^5, the x² coefficient is 10k². If it equals 90, then k² = 9 and k = 3 or −3. Both meet the given squared-coefficient condition; a positive-root assumption would lose one. The x coefficient is 5k, so additional information that it is 15 selects k = 3, while −15 selects k = −3.
A coefficient equation describes a constant multiplier of the chosen power, not a value of x. Substitute candidate parameters into every supplied condition. If two coefficient statements imply incompatible parameter values, state that no parameter satisfies them together. Where a problem supplies a positive or negative restriction, apply it after solving the coefficient equation.
Pause and explain
10k² = 90 has which real parameter values?
Worked example
The coefficient of x² in (1 + kx)^5 is 90. Find all real k. Which remains if the x coefficient is also 15?
Show the worked solution
- The x² coefficient is (5 choose 2)k² = 10k².
- Solve 10k² = 90 to get k = 3 or −3.
- The additional condition 5k = 15 selects k = 3; without it both signs remain valid.
Answer k = 3 or −3 from the x² coefficient; with the x coefficient 15, k = 3 only.
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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 x³ in (2 + x)^5.
Give me a hint
Select k = 3 and keep the remaining power of 2.
Compare my reasoning
- The relevant term is (5 choose 3)2^(5 − 3)x³.
- Compute (5 choose 3) = 10 and 2² = 4.
- Their product is 40, so the term is 40x³ and its coefficient is 40.
40.
Look for these in your work
- Select the term by its variable exponent.
- Include the remaining constant power.
Find the coefficient of x² in (1 + x)(2 − x)^4.
Give me a hint
Use the constant, linear and quadratic terms of the second factor to identify contributions.
Compare my reasoning
- The relevant part of (2 − x)^4 is 16 − 32x + 24x².
- Multiplication by 1 contributes 24x² and multiplication by x contributes −32x².
- Add both contributions to obtain −8x²; higher powers cannot contribute to x² here.
−8.
Look for these in your work
- Find both ways to obtain the target power.
- Add the signed coefficients rather than multiplying them together.
For the model (1 + px)^4, the x coefficient is −8. Find p, the x² and x³ coefficients, and explain whether an x² coefficient of 30 could satisfy the same condition.
Give me a hint
Use the linear coefficient to fix p before checking the other powers.
Compare my reasoning
- 4p = −8 gives p = −2.
- The x² coefficient is 6p² = 24 and the x³ coefficient is 4p³ = −32.
- A proposed x² coefficient of 30 contradicts the already fixed value 24, so no p satisfies those two supplied conditions together.
p = −2; x² coefficient 24; x³ coefficient −32; the proposed coefficient 30 is incompatible.
Look for these in your work
- Use one condition to determine the parameter.
- Verify all remaining conditions instead of solving them independently as if parameters could differ.
Common mistakes
- Match the requested variable power, not just the term number.
- A product can contribute to one power in several ways.
- Do not discard a negative parameter root without a supplied restriction.
What can you explain now?
The x coefficient in (1 + px)^4 is −8. Find p and the x³ coefficient.
Compare with the explanation
p = −2; x³ coefficient −32.
The first condition is 4p = −8, so p = −2; the cubic coefficient is 4p³ = −32.
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Make it stick
Find one binomial coefficient from the general term, then explain how a product or a squared parameter changes the checking steps.
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.