Algebra 2
Polynomial functions
Binomial theorem
20 practice questions
0 video lessons
Theory + worked examples
Theory
The Binomial Theorem expands a power of a binomial:
\[(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^{k},\]
where \(\binom{n}{k}\) is a binomial coefficient — the entries in row \(n\) of Pascal's triangle.
Powers of \(a\) fall from \(n\) to \(0\) while powers of \(b\) rise from \(0\) to \(n\).
Pascal's triangle gives the coefficients.
The binomial theorem.
The expansion:
\[(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^{k}\]
Each term's exponents add to \(n\).
How to expand
- Read the coefficients from row \(n\) of Pascal's triangle.
- Write descending powers of \(a\) and ascending powers of \(b\).
- Multiply in each coefficient.
- Simplify any numerical parts.
Example 1 — Expand a cube
Expand \((a+b)^3\).
Solution
Use row 3 coefficients \(1,3,3,1\).
| \((a+b)^3\) | \(=\) | \(a^3+3a^2b+3ab^2+b^3\) |
Example 2 — With a coefficient
Expand \((x+2)^3\).
Solution
Substitute \(a=x,\ b=2\).
| \(x^3+3x^2(2)+3x(4)+8\) | ||
| \(=\) | \(x^3+6x^2+12x+8\) |
Example 3 — A single term
Find the \(x^2\) term of \((x+3)^4\).
Solution
The term with \(b^2\) uses \(\binom{4}{2}=6\).
| \(\binom{4}{2}x^2 3^2\) | \(=\) | \(6\cdot 9\,x^2\) |
| \(=\) | \(54x^2\) |
Example 4 — Row of coefficients
What are the coefficients of \((a+b)^4\)?
Solution
Read row 4 of Pascal's triangle.
| \(1,\ 4,\ 6,\ 4,\ 1\) |
Common pitfalls
The exponents in each term add to \(n\).
Raise the whole term to its power, including coefficients: \((2)^2=4\).
Row \(n\) has \(n+1\) entries, so there are \(n+1\) terms.
Frequently asked questions
What is the Binomial Theorem?
A formula expanding \((a+b)^n\) using binomial coefficients.
Where do the coefficients come from?
Row \(n\) of Pascal's triangle, or \(\binom{n}{k}\).
How do the exponents behave?
Powers of \(a\) decrease while powers of \(b\) increase; each term's exponents add to \(n\).
How many terms are in the expansion?
\(n+1\) terms for \((a+b)^n\).
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