数学の公式・定理集｜式と証明

$$\begin{align*} &\bullet \ {\left( a+b \right)}^{3} \\[ 7pt ] &\quad = a^{3} + 3{a}^{2}b + 3a{b}^{2} + b^{3} \\[ 10pt ] &\bullet \ {\left( a-b \right)}^{3} \\[ 7pt ] &\quad = a^{3} – 3{a}^{2}b + 3a{b}^{2} – b^{3} \\[ 10pt ] &\bullet \ {\left( a+b \right)}{\left( a^{2}-ab+b^{2} \right)} = a^{3} + b^{3} \\[ 7pt ] &\bullet \ {\left( a-b \right)}{\left( a^{2}+ab+b^{2} \right)} = a^{3} – b^{3} \end{align*}$$

参考

$$\begin{align*} &a^{3} +b^{3} +c^{3} -3abc \\[ 7pt ] &= {\left( a+b+c \right)}{\left( a^{2} +b^{2} +c^{2} -ab -bc-ca \right)} \end{align*}$$

$$\begin{align*} &{\left( a+b \right)}^{n} \\[ 7pt ] &= {}_n \mathrm{ C }_0 a^{n} + {}_n \mathrm{ C }_1 a^{n-1} b + {}_n \mathrm{ C }_2 a^{n-2} b^{2} + \cdots \\[ 15pt ] &\cdots + {}_n \mathrm{ C }_r a^{n-r} b^{r} + \cdots + {}_n \mathrm{ C }_{n-1} a b^{n-1} + {}_n \mathrm{ C }_n b^{n} \end{align*}$$

一般項（第 r+1 項）

$$\begin{equation*} \quad {}_n \mathrm{ C }_r a^{n-r} b^{r} \end{equation*}$$