Binomial Expansion

Prerequist Materials:

Pascal's Triangle

Permutations and Combinations

Expanding Brackets:

$${(a+b)}^0=1$$ $${(a+b)}^1=a+b$$ $${(a+b)}^2=a^2+2ab+b^2$$ $${(a+b)}^3=a^3+3a^{2}b+3a{b}^2+b^3$$ $${(a+b)}^4=a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4$$

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From these expansions we can see that the coefficients of each term are the elements of Pascal's Triangle, and the powers of $a$ count down to one from left to right, whereas, the powers of $b$ count up from one. So...

$${(a+b)}^n=\left(\begin{array}{c}n\\ 0\end{array}\right)a^n+\left(\begin{array}{c}n\\ 1\end{array}\right)a^{n-1}b+\left(\begin{array}{c}n\\ 2\end{array}\right)a^{n-2}{b}^2+\dots +\left(\begin{array}{c}n\\ n-2\end{array}\right)a^2{b}^{n-2}+\left(\begin{array}{c}n\\ n-1\end{array}\right)a{b}^{n-1}+\left(\begin{array}{c}n\\ n\end{array}\right)b^n$$

Worksheets

binomial expansion

$${(1+x)}^n=1+nx+\frac{n(n-1)x^2}{\mathrm{2!}}+\frac{n(n-1)(n-2)x^3}{\mathrm{3!}}+\mathrm{...}$$

where $n\in ℝ$.