31043中３・多項式の展開・計算問題・展開の公式3-3

2023年3月28日 2023年7月19日

banno

次の式を展開しなさい。

(1) $(3 x + y) (3 x - y)$

(2) $(6 x - 7 y) (6 x + 7 y)$

(3) $(8 a - 2 b) (2 b + 8 a)$

(4) $(- 2 x - 5 y) (5 y - 2 x)$

(5) $(- 0.3 a + 0.3 b) (0.3 b + 0.3 a)$

(6) $(\frac{5}{6} x + \frac{3}{4} y) (\frac{5}{6} x - \frac{3}{4} y)$

解答・解説

公式４ $(x + a) (x - a) = x^{2} - a^{2}$

(1) $\begin{array}{r} (3 x + y) (3 x - y) \end{array}$

$\begin{array}{rcl} = & (3 x)^{2} - y^{2} \\ = & 9 x^{2} - y^{2} \end{array}$

答 $9 x^{2} - y^{2}$

(2) $\begin{array}{r} (6 x - 7 y) (6 x + 7 y) \end{array}$

$\begin{array}{rcl} = & (6 x)^{2} - (7 y)^{2} \\ = & 36 x^{2} - 49 y^{2} \end{array}$

答 $36 x^{2} - 49 y^{2}$

(3) $\begin{array}{r} (8 a - 2 b) (2 b + 8 a) \end{array}$

$\begin{array}{rcl} = & (8 a - 2 b) (8 a + 2 b) \\ = & (8 a)^{2} - (2 b)^{2} \\ = & 64 a^{2} - 4 b^{2} \end{array}$

答 $64 a^{2} - 4 b^{2}$

(4) $\begin{array}{r} (- 2 x - 5 y) (5 y - 2 x) \end{array}$

$\begin{array}{rcl} = & (- 2 x - 5 y) (- 2 x + 5 y) \\ = & (- 2 x)^{2} - (5 y)^{2} \\ = & 4 x^{2} - 25 y^{2} \end{array}$

答 $4 x^{2} - 25 y^{2}$

(5) $\begin{array}{r} (- 0.3 a + 0.3 b) (0.3 b + 0.3 a) \end{array}$

$\begin{array}{rcl} = & (0.3 b - 0.3 a) (0.3 b + 0.3 a) \\ = & (0.3 b)^{2} - (0.3 a)^{2} \\ = & 0.09 b^{2} - 0.09 a^{2} \end{array}$

答 $0.09 b^{2} - 0.09 a^{2}$

(6) $\begin{array}{r} (\frac{5}{6} x + \frac{3}{4} y) (\frac{5}{6} x - \frac{3}{4} y) \end{array}$

$\begin{array}{rcl} = & (\frac{5}{6} x)^{2} - (\frac{3}{4} y)^{2} \\ = & \frac{25}{36} x^{2} - \frac{9}{16} y^{2} \end{array}$

答 $\frac{25}{36} x^{2} - \frac{9}{16} y^{2}$