Binomial Theorem
In short: The binomial theorem gives a direct formula for expanding (a + b)ⁿ into a sum of terms, each built from a binomial coefficient, a power of a and a power of b.
PrecalculusAlgebraA-APR.C.5
What is Binomial Theorem?
The binomial theorem replaces repeated multiplication with a pattern. Expanding \((a + b)^{12}\) by hand would mean multiplying twelve brackets together, but the theorem hands you every term at once.
For a whole number \(n\), \((a + b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\). The coefficient \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) counts the ways to pick \(k\) factors that contribute \(b\). The expansion has \(n + 1\) terms, the exponents of \(a\) fall from \(n\) to 0 while those of \(b\) rise from 0 to \(n\), and the coefficients are symmetric.
How to spot it
- A binomial such as (x + 2) or (2x – 3) is raised to a power like 4, 5 or 8, the typical setting for the binomial theorem.
- A question asks for a single term or coefficient of a power, without expanding everything.
- Pascal’s triangle or binomial coefficients like C(8, 3) appear in the problem.
- Probability with repeated trials needs the coefficients of an expansion.
The method
- To apply the binomial theorem, identify \(a\), \(b\) and \(n\) in \((a + b)^n\); treat a subtraction as adding a negative.
- Get the coefficients \(\binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}\) from a factorial formula or Pascal’s triangle.
- Pair each coefficient with \(a^{n-k}\) and \(b^k\), keeping parentheses around coefficients like \(2x\) and \(-3\).
- Simplify each term and test the whole expansion at a convenient value such as \(x = 1\).
Binomial Theorem: solved example
Problem
Expand \((2x – 3)^4\).
Solution
Use \(a = 2x\), \(b = -3\), \(n = 4\) and coefficients 1, 4, 6, 4, 1: \((2x)^4 + 4(2x)^3(-3) + 6(2x)^2(-3)^2 + 4(2x)(-3)^3 + (-3)^4\).
Simplify: \(16x^4 – 96x^3 + 216x^2 – 216x + 81\). Check at \(x = 1\): \(16 – 96 + 216 – 216 + 81 = 1 = (-1)^4\). The expansion is \(16x^4 – 96x^3 + 216x^2 – 216x + 81\).
Try it
Question 1
Find the coefficient of \(x^3\) in \((x + 2)^5\).
Show the answer
\(\binom{5}{3}\cdot 2^2 = 10 \cdot 4 = 40\).
Question 2
How many terms does \((3a – b)^{12}\) have, and what is the first one?
Show the answer
It has \(12 + 1 = 13\) terms, and the first is \((3a)^{12} = 531{,}441a^{12}\).
Binomial Theorem vs. Pascal's Triangle
The triangle is a table of numbers, each the sum of the two above it. The theorem is the rule that says how to use a row of that table as the coefficients of an expansion. The triangle is handy for small powers; for \((a + b)^{20}\), the factorial formula is quicker.
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