31023中３・多項式の展開・計算問題・展開の公式1-3

(1) $\quad\;\,\begin{eqnarray}(\,2x+y\,)(\,2x+5y\,)\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&(\,2x\,)^{2}+(\,y+5y\,) \times 2x+(\,y\times 5y\,)\\[6pt]&=&4x^{2}+12xy+5y^{2}\end{eqnarray}\;\;$

答$4x^{2}+12xy+5y^{2}$

(2) $\quad\;\,\begin{eqnarray}(\,4x-3y\,)(\,4x-7y\,)\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&(\,4x\,)^{2}+\{\,(\,-3y\,)+(\,-7y\,)\} \times 4x+ \{(\,-3y\,) \times (\,-7y\,)\}\\[6pt]&=&16x^{2}-40xy+21y^{2}\end{eqnarray}\;\;$

答$16x^{2}-40xy+21y^{2}$

(3) $\quad\;\,\begin{eqnarray}(\,6x+5y\,)(\,6x-4y\,)\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&(\,6x\,)^{2}+\{\,5y+(-4y)\} \times 6x+\{\,5y \times (\,-4y\,)\}\\[6pt]&=&36x^{2}+6xy-20y^{2}\end{eqnarray}\;\;$

答$36x^{2}+6xy-20y^{2}$

(4) $\quad\;\,\begin{eqnarray}(\,-3x-7y\,)(\,-3x+8y\,)\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&(\,-3x\,)^{2}+\{(\,-7y\,)+8y\,\} \times (\,-3x\,)+\{(\,-7y\,) \times 8y\,\}\\[6pt]&=&9x^{2}-3xy-56y^{2}\end{eqnarray}\;\;$

《 別解 》

$\quad\;\;\;\begin{eqnarray}(\,-3x-7y\,)(\,-3x+8y\,)\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&\{-(\,3x+7y\,)\}\{-(\,3x-8y\,)\}\\[6pt]&=&(\,3x+7y\,)(\,3x-8y\,)\\[6pt]&=&(\,3x\,)^{2}+\{\,7y+(\,-8y\,)\} \times 3x +\{\,7y \times (\,-8y\,)\}\\[6pt]&=&9x^{2}-3xy-56y^{2}\end{eqnarray}\;\;$

答$9x^{2}-3xy-56y^{2}$

(5) $\quad\;\,\begin{eqnarray}(\,2x-3y\,)(\,5x-3y\,)\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&(\,-3y+2x\,)(\,-3y+5x\,)\\[6pt]&=&(\,-3y\,)^{2}+(\,2x+5x\,) \times (\,-3y\,)+(\,2x \times 5x\,)\\[6pt]&=&9y^{2}-21xy+10x^{2}\end{eqnarray}\;\;$

《 別解 》

$\quad\;\;\;\begin{eqnarray}(\,2x-3y\,)(\,5x-3y\,)\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&(\,-3y+2x\,)(\,-3y+5x\,)\\[6pt]&=&\{-(\,3y-2x\,)\}\{-(\,3y-5x\,)\}\\[6pt]&=&(\,3y-2x\,)(\,3y-5x\,)\\[6pt]&=&(\,3y\,)^{2}+\{(\,-2x\,)+(\,-5x\,)\} \times 4x+\{(\,-2x\,) \times (\,-5x\,)\}\\[6pt]&=&9y^{2}-21xy+10x^{2}\end{eqnarray}\;\;$

答$9y^{2}-21xy+10x^{2}$