Binomial Theorem — Formula, Expansion & Examples

The binomial theorem is a procedure for expanding expressions like $(a+b)^n$ without multiplying the brackets out one at a time. In the standard algebra-class version, it applies when $n$ is a non-negative integer:

The formula hands you the coefficient of each term and shows how the powers of $a$ and $b$ change across the expansion.

When to use this method

Use the binomial theorem when you need a full polynomial expansion, one specific term in an expansion, or a quick way to read off coefficients without repeated multiplication. It appears in algebra first, then later in probability, series, and some calculus approximations. The condition is that $n$ is a non-negative integer; if the exponent is not, this finite formula no longer produces a polynomial and you need a different version of the idea.

Step by step

1. Write each term in the standard form $\binom{n}{k} a^{n-k} b^k$, with $k$ running from $0$ to $n$.
2. Read the coefficients $\binom{n}{k}$. The power of $a$ drops from $n$ to $0$, and the power of $b$ rises from $0$ to $n$. In every term the exponents add up to $n$.
3. Substitute the actual values of $a$ and $b$ into each term, keeping signs attached.
4. Simplify term by term.

For example, when $n=4$, the coefficients are $1,\ 4,\ 6,\ 4,\ 1$ — the same numbers as the $4$th row of Pascal's triangle. They arise because expanding $(a+b)(a+b)\cdots(a+b)$ means choosing one term from each of $n$ identical brackets; to land $b^k$ you pick $b$ from exactly $k$ brackets and $a$ from the rest, which can be done in $\binom{n}{k}$ ways. That count is the coefficient, which is also why the middle coefficients are usually the largest.

A full worked example: $(2x-3)^4$

Because the exponent is $4$, the theorem applies directly. The coefficients are

Using the general pattern,

Now simplify term by term:

So the expansion is

Two quick checks confirm it: the exponents in each term add to $4$, and the signs alternate correctly because the second part is $-3$.

Where students get stuck, and how to check each step

• Step 1, the whole-expansion shortcut: The most common error is treating $(a+b)^n$ as $a^n + b^n$. That is false in general because the middle terms matter.
• Step 2, matching coefficients to powers: Using the right coefficients with the wrong powers breaks the expansion. Self-check: do the exponents on the two parts add up to $n$ in every term?
• Step 3, signs: In $(2x-3)^4$ the second part is $-3$, not $3$, so odd powers stay negative and even powers turn positive. Keep the sign attached when you substitute.

To run the procedure yourself, expand $(x+2)^5$. Before simplifying, verify the coefficients are $1, 5, 10, 10, 5, 1$ and that the exponents add to $5$ in every term. Then compare your middle terms first, since that is where coefficient and sign errors surface fastest.