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Pre-Calculus Sequences and series (advanced)

Binomial theorem (general expansion)

20 practice questions 0 video lessons Theory + worked examples

The Binomial Theorem

Common Core Pre-Calculus • Standard A-APR.5 • Sequences & Series

The Binomial Theorem is a topic in Sequences & Series in the Common Core State Standards. It is aligned to Standard A-APR.5, which requires students to use the Binomial Theorem to expand a power of a binomial.

The Binomial Theorem expands \((a+b)^n\) using the binomial coefficients \(\dbinom{n}{k}\), which are the entries of Pascal's triangle.

Common Core Pre-Calculus › Sequences & Series › The Binomial Theorem  —  Standard A-APR.5

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Theory

The Binomial Theorem expands a power of a binomial without multiplying it out term by term:

\[(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{\,n-k}b^{\,k},\]

where the binomial coefficients are \(\displaystyle\binom{n}{k}=\dfrac{n!}{k!\,(n-k)!}\). These are exactly the entries of Pascal's triangle.

Across the expansion the power of \(a\) decreases from \(n\) to \(0\) while the power of \(b\) increases from \(0\) to \(n\).

Pascal's triangle is the quick route for small \(n\): each row lists the coefficients, and each entry is the sum of the two above.
Pascal's triangle Pascal's triangle gives the binomial coefficients; each entry is the sum of the two above it. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1rows give the coefficients of (a+b)ⁿ
Pascal's triangle supplies the coefficients of \((a+b)^n\).
Binomial Theorem Binomial Theorem Binomial Theorem (a+b)ⁿ = ∑ C(n,k) aⁿ₋ᵤ bᵤ C(n,k) = n!k!(n−k)!
The Binomial Theorem and the coefficient formula.

The theorem and the general term:

\[(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{\,n-k}b^{\,k}\]
\[\text{term for a given }k:\ \binom{n}{k}a^{\,n-k}b^{\,k}\]
a plus b to the n is the sum of n choose k times a to the n minus k times b to the k
To target one term, pick the \(k\) that gives the power you want — no full expansion needed.

How to expand a binomial

  1. Get the coefficients from row \(n\) of Pascal's triangle (or \(\binom{n}{k}\)).
  2. Write the powers: \(a\) from \(n\) down to \(0\), \(b\) from \(0\) up to \(n\).
  3. Combine coefficient and powers for each term.
  4. For a single term, solve for the \(k\) giving the desired power.
Example 1 — Expand with Pascal's triangle
Expand \((a+b)^3\).
Solution

Row 3 of Pascal's triangle is \(1,3,3,1\); decrease the power of \(a\), increase \(b\).

\[a^3+3a^2b+3ab^2+b^3\]
expansion is a cubed plus 3 a squared b plus 3 a b squared plus b cubed
Example 2 — With numbers
Expand \((x+2)^3\).
Solution

Use coefficients \(1,3,3,1\) with \(b=2\).

\(=\)x^3+3x^2(2)+3x(2)^2+(2)^3
\(=\)x^3+6x^2+12x+8
expansion is x cubed plus 6 x squared plus 12 x plus 8
Example 3 — A single term
Find the \(x^2\) term of \((x+1)^5\).
Solution

The general term is \(\binom{5}{k}x^{5-k}(1)^k\); for \(x^2\), \(5-k=2\Rightarrow k=3\).

\binom{5}{3}x^2\(=\)10x^2
the x squared term is 10 x squared
Example 4 — A binomial coefficient
Evaluate \(\binom{6}{2}\).
Solution

Use \(\binom{n}{k}=\dfrac{n!}{k!(n-k)!}\).

\binom{6}{2}\(=\)\dfrac{6!}{2!\,4!}
\(=\)\dfrac{6\cdot 5}{2}=15
6 choose 2 is 15

Common pitfalls

Powers sum to \(n\) in every term. \(a^{\,n-k}b^{\,k}\) always has total degree \(n\).
Include the coefficient's own powers. In \((x+2)^n\), the \(2\) is raised too.
\(\binom{n}{k}\) counts from \(k=0\). Row \(n\) has \(n+1\) entries.

Frequently asked questions

What is the Binomial Theorem?

A formula for expanding \((a+b)^n\): \(\sum_{k=0}^{n}\binom{n}{k}a^{\,n-k}b^{\,k}\).

What is a binomial coefficient?

\(\binom{n}{k}=\dfrac{n!}{k!(n-k)!}\), the number of ways to choose \(k\) items from \(n\); it is the entry in Pascal's triangle.

How does Pascal's triangle help?

Each row gives the coefficients of \((a+b)^n\), and each entry is the sum of the two directly above it.

How do you find a single term of an expansion?

Use \(\binom{n}{k}a^{\,n-k}b^{\,k}\) and pick the \(k\) that produces the power you want.

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