AP® Precalculus · Unit 1 · Topic 1.11
The Binomial Theorem
The Binomial Theorem gives the full expansion of (a+b)n without multiplying it out by hand, using combinations C(n,k) as coefficients that match Pascal's triangle.
Expand (a+b)4 by brute force and you get coefficients 1, 4, 6, 4, 1. Those are exactly row 4 of Pascal's triangle. That is not a coincidence: the Binomial Theorem says (a+b)n = sum of C(n,k) an-k bk for k = 0 to n, where C(n,k) counts how many ways k of the n factors contribute a b instead of an a. Pascal's triangle builds each entry by adding the two entries above it, because choosing k items from n either includes a specific item or leaves it out, splitting into C(n-1,k-1) + C(n-1,k) = C(n,k). That recursive rule is why the triangle and the theorem always agree.
Every expansion follows the same pattern: powers of the first term start at n and count down, powers of the second term start at 0 and count up, and the two exponents in each term always add to n. The coefficients come straight from the matching row of the triangle, or from computing C(n,k) directly. When the second term is negative, like in (x-2)5, treat it as (x + (-2))5 and let the sign of (-2)k do the work: odd k gives a negative term, even k gives a positive one. The magnitudes never change, only the signs alternate.
This theorem is the shortcut version of repeated polynomial multiplication, and it connects directly to how polynomials behave near zero versus far away. Knowing the expansion pattern also makes it easy to pull out a single term, like the x2 coefficient, without expanding the whole thing, a skill that shows up again when working with series and rational function behavior.
Key terms
- Binomial Theorem
- A formula giving the full expansion of (a+b)n as a sum of terms C(n,k) an-k bk.
- Pascal's triangle
- A triangular array where each entry is the sum of the two entries above it, giving binomial coefficients row by row.
- Binomial coefficient C(n,k)
- The number of ways to choose k items from n, equal to n! / (k!(n-k)!).
- Term
- One piece of the expansion of the form C(n,k) an-k bk, with exponents always adding to n.
Watch it explained
- How it worksHow does each entry of Pascal's triangle come from adding the two entries resting on it?
- Worked exampleShow me (x + 3)⁴ expanded by the binomial theorem, with every coefficient traced to its source.
- Going furtherWhat happens to the expansion when the second term is negative, as in (x - 2)⁵?
Check yourself
End of topic.