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# 特殊的二项式乘积回顾

"平方差" 模式:
$\left(a+b\right)\left(a-b\right)={a}^{2}-{b}^{2}$

$\begin{array}{rl}& \left(a+b{\right)}^{2}={a}^{2}+2ab+{b}^{2}\\ \\ & \left(a-b{\right)}^{2}={a}^{2}-2ab+{b}^{2}\end{array}$

### 例题1

$\left(c-5\right)\left(c+5\right)$

$\left(a+b\right)\left(a-b\right)={a}^{2}-{b}^{2}$

$\left(c-5\right)\left(c+5\right)={c}^{2}-25$

$\begin{array}{rl}& \left(c-5\right)\left(c+5\right)\\ \\ =& c\left(c\right)+c\left(5\right)-5\left(c\right)-5\left(5\right)\\ \\ =& c\left(c\right)+5c-5c-5\left(5\right)\\ \\ =& {c}^{2}-25\end{array}$

### 例题 2

$\left(m+7{\right)}^{2}$

$\left(a+b{\right)}^{2}={a}^{2}+2ab+{b}^{2}$

$\left(m+7{\right)}^{2}={m}^{2}+14m+49$

$\begin{array}{rl}& \left(m+7{\right)}^{2}\\ \\ =& \left(m+7\right)\left(m+7\right)\\ \\ =& m\left(m\right)+m\left(7\right)+7\left(m\right)+7\left(7\right)\\ \\ =& m\left(m\right)+7m+7m+7\left(7\right)\\ \\ =& {m}^{2}+14m+49\end{array}$

### 例题 3

$\left(6w-y\right)\left(6w+y\right)$

$\left(a+b\right)\left(a-b\right)={a}^{2}-{b}^{2}$

$\begin{array}{rl}& \left(6w-y\right)\left(6w+y\right)\\ \\ =& \left(6w{\right)}^{2}-{y}^{2}\\ \\ =& 36{w}^{2}-{y}^{2}\end{array}$

$\begin{array}{rl}& \left(6w-y\right)\left(6w+y\right)\\ \\ =& 6w\left(6w\right)+6w\left(y\right)-y\left(6w\right)-y\left(y\right)\\ \\ =& 6w\left(6w\right)+6wy-6wy-y\left(y\right)\\ \\ =& 36{w}^{2}-{y}^{2}\end{array}$