32052中３・多項式の因数分解・計算問題・いろいろな因数分解2

(1) $\quad\;\,\begin{eqnarray}ax^{2}-10ax+24a\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&a(\,x^{2}-10x+24\,)\\[6pt]&=&a\{\,x^{2}+(-4-6)x+(-4) \times (-6)\,\}\\[6pt]&=&a(\,x-4\,)(\,x-6\,)\end{eqnarray}\;\;$

答$a(\,x-4\,)(\,x-6\,)$

(2) $\quad\;\,\begin{eqnarray}3mx^{2}+12mx-36m\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=& 3m(\,x^{2}+4x-12\,) \\[6pt]&=&3m\{\,x^{2}+(-2+6)x+(-2) \times 6\,\}\\[6pt]&=&3m(\,x-2\,)(\,x+6\,)\end{eqnarray}\;\;$

答$3m(\,x-2\,)(\,x+6\,)$

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

$\;\;\begin{eqnarray}&=&a(\,a^{2}-2a+1\,) \\[6pt]&=&a(\,a^{2}-2 \times 1 \times a + 1^{2}\,)\\[6pt]&=&a(\,a-1\,)^{2}\end{eqnarray}\;\;$

答$a(\,a-1\,)^{2}$

(4) $\quad\;\,\begin{eqnarray}-8a^{2}m+24abm-18b^{2}m\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&-2m(\,4a^{2}-12ab+9b^{2}\,) \\[6pt]&=& -2m\{\,(2a)^{2}-2 \times 3b \times 2a+ (3b)^{2}\} \\[6pt]&=&-2m(\,2a-3b\,)^{2}\end{eqnarray}\;\;$

答$-2m(\,2a-3b\,)^{2}$

(5) $\quad\;\,\begin{eqnarray}16ab^{2}-36a^{3}\end{eqnarray}\;\;$

$\;\;\begin{eqnarray}&=&4a(\,4b^{2}-9a^{2}\,) \\[6pt]&=&4a\{(\,2b\,)^{2}-(\,3a\,)^{2}\} \\[6pt]&=&4a(\,2b+3a\,)(\,2b-3a\,)\end{eqnarray}\;\;$

答$4a(\,2b+3a\,)(\,2b-3a\,)$