Don't expand everything — grab one term: To find just one term of (a + b)ⁿ, use the general term: choose r of the b's. Match the power of x you want to fix r, then compute.
IB-style question — coefficient of one power
Find the coefficient of x³ in the expansion of (2 + x)⁵.
Step by step
- General term: choose r of the x's. The power of x is r, so for x³ take r = 3.
- Substitute r = 3.
- Read off the coefficient.
Final answer
40.
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Write the coefficient, set it equal, solve: If a coefficient is given a value, write that coefficient as an expression in the unknown, set it equal to the number, and solve.
IB-style question — find the unknown
In the expansion of (1 + ax)⁵, the coefficient of x² is 40.
Given that a > 0, find the value of a.
Step by step
- General term: ⁵Cᵣ (ax)ʳ. For x² take r = 2.
- So the coefficient of x² is 10a². Set it equal to 40.
- a > 0, so take the positive root.
Final answer
a = 2.