Give the expansion of the binomial. Simplify your answer.

a. $(a+b)^3$

b. $(20+1)^3$

c. $(20-1)^3$

Solutions

VerifiedThe binomial expansion of $a$ and $b$ is given by:

$\begin{align*} (a+b)^n &= (_nC_0)(a^n)(b^0) + (_nC_1)(a^{n-1})(b^1) + (_nC_2)(a^{n-2})(b^2) +\\ & (_nC_3)(a^{n-3})(b^3) + \text{ . . . } + (_nC_n)(a^0)(b^n). \end{align*}$

$(a+b)^{n}={}_{n}C_{0}a^{n}b^{0}+{}_{n}C_{1}a^{n-1}b^{1}+{}_{n}C_{2}a^{n-2}b^{2}+...+{}_{n}C_{n}a^{0}b^{n}.\\\\$ The coefficients can be found in row 3 of Pascal's triangle:\\$

$\begin{array}{llllllllllllllllll} n & & & & & & & & & 1 & & & & & & & & \\ & & & & & & & & & & & & & & & & & \\ 2 & & & & & & & 1 & & 2 & & 1 & & & & & & \\ 3 & & & & & & 1 & & 3 & & 3 & & 1 & & & & & \end{array}$

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