SL 1.9—Binomial theorem

Expand a power with binomial coefficients.

For a non-negative integer n, (a+b)ⁿ is the sum of terms whose coefficients are nCr and whose powers of a and b add to n.

Worked example
(x+2)²=x²+4x+4; coefficients 1,2,1 come from Pascal’s triangle.

Worked example
What should the powers add to? n in every term.

Common boundary
Do not omit the middle term or change the coefficient pattern.

Full binomial theorem: $(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r$ for $n\in\mathbb{N}$, where $\binom{n}{r}=n!/[r!(n-r)!]$. Example: $(2x-1)^3=\binom30(2x)^3+\binom31(2x)^2(-1)+\binom32(2x)(-1)^2+\binom33(-1)^3=8x^3-12x^2+6x-1$. Each term's total power is $n$.