31015中３・多項式の展開・計算問題・分配法則5

(1) $\;\;\begin{eqnarray}(a+2)(a+b+3)\end{eqnarray}\;\;$

$\qquad$ 解法１

$\quad\qquad\begin{eqnarray}(a+2)(a+b+3) \end{eqnarray}\;\;$

$\qquad\begin{eqnarray}&=&a(a+b+3)+2(a+b+3)\\[3pt]&=&a^{2}+ab+3a+2a+2b+6\\[3pt]&=&a^{2}+ab+5a+2b+6\end{eqnarray}\;\;$

$\qquad$ 解法２

$\quad\qquad\begin{eqnarray}(a+2)(a+b+3)\end{eqnarray}\;\;$

$\qquad\begin{eqnarray} &=&a\times a+a\times b+a\times 3+2\times a+2\times b+2\times 3\\[3pt] &=&a^{2}+ab+3a+2a+2b+6\\[3pt]&=&a^{2}+ab+5a+2b+6\end{eqnarray}\;\;$

答$a^{2}+ab+5a+2b+6$

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

$\qquad$ 解法１

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

$\qquad\begin{eqnarray}&=&x(2x+3y-5)-y(2x+3y-5)\\[3pt]&=&2x^{2}+3xy-5x-2xy-3y^{2}+5y\\[3pt]&=&2x^{2}+xy-5x-3y^{2}+5y\end{eqnarray}\;\;$

$\qquad$ 解法２

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

$\qquad\begin{eqnarray}&=&x \hspace{-0.2em} \times \hspace{-0.2em} 2x+x \hspace{-0.2em} \times \hspace{-0.2em} 3y+x \hspace{-0.2em} \times \hspace{-0.2em} (-5)-y \hspace{-0.2em} \times \hspace{-0.2em} 2x-y \hspace{-0.2em} \times \hspace{-0.2em} 3y-y \hspace{-0.2em} \times \hspace{-0.2em} (-5)\\[3pt]&=&2x^{2}+3xy-5x-2xy-3y^{2}+5y\\[3pt]&=&2x^{2}+xy-5x-3y^{2}+5y\end{eqnarray}\;\;$

答$2x^{2}+xy-5x-3y^{2}+5y$

(3) $\;\;\begin{eqnarray}(3a-b-8)(2a-3b)\end{eqnarray}\;\;$

$\qquad$ 解法１

$\quad\qquad\begin{eqnarray}(3a-b-8)(2a-3b)\end{eqnarray}\;\;$

$\qquad\begin{eqnarray}&=&3a(2a-3b)-b(2a-3b)-8(2a-3b)\\[3pt]&=&6a^{2}-9ab-2ab+3b^{2}-16a+24b\\[3pt]&=&6a^{2}-11ab+3b^{2}-16a+24b\end{eqnarray}\;\;$

$\qquad$ 解法２

$\quad\qquad\begin{eqnarray}(3a-b-8)(2a-3b)\end{eqnarray}\;\;$

$\qquad\begin{eqnarray}&=&3a\hspace{-0.3em}\times \hspace{-0.3em} 2a+3a \hspace{-0.3em} \times \hspace{-0.3em} (-3b)-b \hspace{-0.3em} \times \hspace{-0.3em} 2a-b \hspace{-0.3em} \times \hspace{-0.3em} (-3b)-8 \hspace{-0.3em} \times \hspace{-0.3em} 2a-8 \hspace{-0.3em} \times \hspace{-0.3em} (-3b) \\[3pt]&=&6a^{2}-9ab-2ab+3b^{2}-16a+24b\\[3pt]&=&6a^{2}-11ab+3b^{2}-16a+24b+6\end{eqnarray}\;\;$

答$6a^{2}-11ab+3b^{2}-16a+24b$