Binomial Theorem

Multiplying $(a + b)^2$ by hand is no big deal — you get $a^2 + 2ab + b^2$. But what about $(a + b)^{10}$? That would be a nightmare to multiply out the long way. The Binomial Theorem is a shortcut that tells you exactly what every term will be without doing all that multiplication. It's like a recipe: just follow the pattern and you get the answer.

The pattern is built on Pascal's Triangle — a triangle of numbers where each number is the sum of the two numbers above it. Row 0 is just 1. Row 1 is 1, 1. Row 2 is 1, 2, 1. Row 4 is 1, 4, 6, 4, 1. These numbers become the coefficients (the multipliers) in your expansion. The triangle also gives you the binomial coefficients $\binom{n}{k}$, which count how many ways to choose $k$ items from $n$ — but for this class, just think of them as the numbers in Pascal's Triangle.

Here's the pattern for each term: as you go left to right, $a$'s exponent counts down from $n$ to 0, while $b$'s exponent counts up from 0 to $n$. They always add up to $n$. So in $(x + 2)^3$, the terms involve $x^3$, then $x^2 \cdot 2$, then $x \cdot 2^2$, then $2^3$, with Pascal's Triangle coefficients 1, 3, 3, 1 in front.

The biggest trap is negative signs. If your expression is $(x - 3)^4$, treat it as $(x + (-3))^4$ so that $b = -3$. The negative sign gets handled automatically by the powers — odd powers of $-3$ are negative, even powers are positive, so the signs alternate. Also, if $a$ is something like $2x$, you must raise all of it to the power: $(2x)^3 = 8x^3$, not $2x^3$.