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# 特殊的二项式乘积回顾

"平方差" 模式:
left parenthesis, a, plus, b, right parenthesis, left parenthesis, a, minus, b, right parenthesis, equals, a, squared, minus, b, squared

\begin{aligned} &(a+b)^2=a^2+2ab+b^2\\\\ &(a-b)^2=a^2-2ab+b^2 \end{aligned}

### 例题1

left parenthesis, c, minus, 5, right parenthesis, left parenthesis, c, plus, 5, right parenthesis

left parenthesis, a, plus, b, right parenthesis, left parenthesis, a, minus, b, right parenthesis, equals, a, squared, minus, b, squared

left parenthesis, c, minus, 5, right parenthesis, left parenthesis, c, plus, 5, right parenthesis, equals, c, squared, minus, 25

\begin{aligned} &(\purpleD{c-5})(c+5)\\\\ =&\purpleD{c}(c)+\purpleD{c}(5)\purpleD{-5}(c)\purpleD{-5}(5)\\\\ =&\purpleD{c}(c)+\redD{5c-5c}\purpleD{-5}(5)\\\\ =&c^2-25 \end{aligned}

### 例题 2

left parenthesis, m, plus, 7, right parenthesis, squared

left parenthesis, a, plus, b, right parenthesis, squared, equals, a, squared, plus, 2, a, b, plus, b, squared

left parenthesis, m, plus, 7, right parenthesis, squared, equals, m, squared, plus, 14, m, plus, 49

\begin{aligned} &(m+7)^2\\\\ =&(\blueD{m+7})(m+7)\\\\ =&\blueD{m}(m)+\blueD{m}(7)+\blueD{7}(m)+\blueD{7}(7)\\\\ =&\blueD{m}(m)\greenD{+7m+7m}+\blueD{7}(7)\\\\ =&m^2+14m+49 \end{aligned}

### 例题 3

left parenthesis, 6, w, minus, y, right parenthesis, left parenthesis, 6, w, plus, y, right parenthesis

left parenthesis, a, plus, b, right parenthesis, left parenthesis, a, minus, b, right parenthesis, equals, a, squared, minus, b, squared

\begin{aligned} &(6w-y)(6w+y) \\\\ =&(6w)^2-y^2 \\\\ =&36w^2-y^2 \end{aligned}

\begin{aligned} &(\purpleD{6w-y})(6w+y)\\\\ =&\purpleD{6w}(6w)+\purpleD{6w}(y)\purpleD{-y}(6w)\purpleD{-y}(y)\\\\ =&\purpleD{6w}(6w)+\redD{6wy-6wy}\purpleD{-y}(y)\\\\ =&36w^2-y^2 \end{aligned}