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Hard

Finding the Coefficient of a Specific Term

Simple Explanation

To find the coefficient of a specific power of x in a binomial expansion (like the coefficient of x⁵), set up the general term formula, solve for the value of r that gives that exact power of x, and then evaluate only the numeric coefficient at that r.

Why Do We Need It?

This is the most common practical use of the binomial theorem — questions rarely ask for a "term number," but frequently ask for the coefficient of a specific power of a variable.

Formula

The General Term of a Binomial Expansion

T(r+1) = ⁿCᵣ · aⁿ⁻ʳ · bʳ

Gives the formula for the (r+1)th term of the expansion of (a+b)ⁿ directly, without writing out every earlier term first.

T(r+1)
— the (r+1)th term of the expansion (the "+1" exists because the first term corresponds to r=0)
n, r
— as in the binomial theorem — n is the power, r is the term index

When to use it: Whenever you need one specific term of a binomial expansion (e.g. the 5th term, or the term containing x⁴) without expanding the whole thing.

Worked Example

Find the coefficient of a specific power

Find the coefficient of x⁴ in the expansion of (x + 3)⁶.

    Why Does This Work?

    Every term in the expansion has a distinct power of x (from x⁰ up to xⁿ), so there is exactly one value of r that produces the requested power — solving "exponent of x = target" for r identifies that one term uniquely, and the general term formula then gives its exact coefficient.

    Real-Life Example

    Approximating (1+x)ⁿ for small x

    Scientists often approximate expressions like (1+x)ⁿ for small x by keeping only the first few terms, needing the exact coefficient of each low power of x.

    Finding the coefficient of x², x³, etc. this way builds an accurate polynomial approximation used throughout physics and engineering.

    Practice

    Find the coefficient of x³ in the expansion of (x + 2)⁵.

    Hard

    Common mistake

    Forgetting to include the power of b in the coefficient — the "coefficient of x⁴" is ⁿCᵣ × bʳ together, not just the binomial coefficient ⁿCᵣ by itself.

    Quick Review

    • Set the exponent of x in the general term equal to the target power, and solve for r.
    • Evaluate ⁿCᵣ × bʳ at that r to get the full coefficient — not just ⁿCᵣ alone.
    • A very common practical question format for the binomial theorem.
    ← Back to The Binomial Theorem