32023中３・多項式の因数分解・計算問題・因数分解の公式1-3

(1) $\quad\;\,\begin{eqnarray}x^{2}+5xy+6y^{2}\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&x^{2}+ \color{red}\{(+2y)+(+3y)\} \color{black}x+\color{red}\{(+2y) \times (+3y)\} \\[6pt]&=&(\,x+2y\,)(\,x+3y\,)\end{eqnarray}\;\;$

答$(\,x+2y\,)(\,x+3y\,)$

(2) $\quad\;\,\begin{eqnarray}x^{2}-9xy+20y^{2}\end{eqnarray}\;\;$

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

答$(\,x-4y\,)(\,x-5y\,)$

(3) $\quad\;\,\begin{eqnarray}a^{2}-30ab+200b^{2}\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&a^{2}+ \color{red}\{(-10b)+(-20b)\} \color{black}a+\color{red}\{(-10b) \times (-20b)\} \\[6pt]&=&(\,a-10b\,)(\,a-20b\,)\end{eqnarray}\;\;$

答$(\,a-10b\,)(\,a-20b\,)$

(4) $\quad\;\,\begin{eqnarray}x^{2}+xy-30y^{2}\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&x^{2}+ \color{red}\{(+6y)+(-5y)\} \color{black}x+ \color{red}\{(+6y) \times (-5y)\} \\[6pt]&=&(\,x+6y\,)(\,x-5y\,)\end{eqnarray}\;\;$

答$(\,x+6y\,)(\,x-5y\,)$

(5) $\quad\;\,\begin{eqnarray}x^{2}-2xy-48y^{2}\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&x^{2}+ \color{red}\{(+6y)+(-8y)\} \color{black}x+ \color{red}\{(+6y) \times (-8y)\} \\[6pt]&=&(\,x+6y\,)(\,x-8y\,)\end{eqnarray}\;\;$

答$(\,x+6y\,)(\,x-8y\,)$

(6) $\quad\;\,\begin{eqnarray}a^{2}+5ab-84b^{2}\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&a^{2}+ \color{red}\{(\,+12b\,)+(\,-7b\,)\} \color{black}a+ \color{red}\{(\,+12b\,) \times (\,-7b\,)\}\\[6pt]&=&(\,a+12b\,)(\,a-7b\,)\end{eqnarray}\;\;$

答$(\,a+12b\,)(\,a-7b\,)$