Binomial Theorem — Formula, Expansion & Examples

The binomial theorem gives a fast way to expand expressions like $(a+b)^n$. In the standard version used in algebra class, it applies when $n$ is a non-negative integer.

Instead of multiplying $(a+b)(a+b)(a+b)$ by hand, you can use one pattern for the whole expansion:

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

This formula tells you the coefficient of each term and how the powers of $a$ and $b$ change across the expansion.

Binomial Theorem Formula And Pattern

Each term in the expansion has the form

$$\binom{n}{k} a^{n-k} b^k$$

with $k$ running from $0$ to $n$.

That means the coefficient is $\binom{n}{k}$, the power of $a$ drops from $n$ to $0$, and the power of $b$ rises from $0$ to $n$. In every term, the exponents still add up to $n$.

For example, when $n=4$, the coefficients are

$$1,\ 4,\ 6,\ 4,\ 1$$

These are the same numbers as the $4$th row of Pascal's triangle.

Why The Coefficients Work

If you expand $(a+b)^n$, you are really choosing one term from each of $n$ identical brackets:

$$(a+b)(a+b)\cdots(a+b)$$

To get a term with $b^k$, you must choose $b$ from exactly $k$ brackets and choose $a$ from the rest. The number of ways to do that is $\binom{n}{k}$, so that count becomes the coefficient.

This is also why the middle coefficients are usually the largest: there are more ways to split the choices near the middle than at the ends.

Binomial Expansion Example: $(2x-3)^4$

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

$$1,\ 4,\ 6,\ 4,\ 1$$

Using the general pattern,

$$(2x-3)^4 = (2x)^4 + 4(2x)^3(-3) + 6(2x)^2(-3)^2 + 4(2x)(-3)^3 + (-3)^4$$

Now simplify term by term:

$$(2x)^4 = 16x^4$$ $$4(2x)^3(-3) = 4(8x^3)(-3) = -96x^3$$ $$6(2x)^2(-3)^2 = 6(4x^2)(9) = 216x^2$$ $$4(2x)(-3)^3 = 4(2x)(-27) = -216x$$ $$(-3)^4 = 81$$

So the expansion is

$$(2x-3)^4 = 16x^4 - 96x^3 + 216x^2 - 216x + 81$$

Notice two quick checks that help catch mistakes fast: the exponents in each term add to $4$, and the signs alternate correctly because the second term is $-3$.

Common Binomial Theorem Mistakes

The most common mistake is treating $(a+b)^n$ as $a^n + b^n$. That is false in general because the middle terms matter.

Another mistake is using the right coefficients with the wrong powers. In every term, the exponents on the two parts should add up to the original exponent $n$.

Negative signs also cause trouble. In $(2x-3)^4$, the second term is $-3$, not just $3$, so odd powers of that factor stay negative and even powers become positive.

When To Use The Binomial Theorem

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 shows up in algebra first, then later in probability, series, and some calculus approximations. If the exponent is not a non-negative integer, this finite formula no longer gives a polynomial, so you need a different version of the idea.

Try A Similar Expansion

Expand $(x+2)^5$ and check two things before you simplify: the coefficients should be $1, 5, 10, 10, 5, 1$, and the exponents on the two parts should add to $5$ in every term.

If you want to go one step further, try your own version in the solver and compare your middle terms first. That is where coefficient and sign errors usually show up fastest.