次の式を展開せよ。 (1) $(a+b)^2(a-b)^2$ (2) $(x^2+1)(x+1)(x-1)$ (3) $(x^2+2x+3)(x^2-2x+3)$ (4) $(x-y+z)(x+y-z)$

1. 問題の内容

次の式を展開せよ。

(1) $(a+b)^2(a-b)^2$

(2) $(x^2+1)(x+1)(x-1)$

(3) $(x^2+2x+3)(x^2-2x+3)$

(4) $(x-y+z)(x+y-z)$

2. 解き方の手順

(1) $(a+b)^2(a-b)^2 = ((a+b)(a-b))^2 = (a^2-b^2)^2 = a^4 - 2a^2b^2 + b^4$

(2) $(x^2+1)(x+1)(x-1) = (x^2+1)(x^2-1) = x^4 - 1$

(3) $(x^2+2x+3)(x^2-2x+3) = ((x^2+3)+2x)((x^2+3)-2x) = (x^2+3)^2 - (2x)^2 = x^4 + 6x^2 + 9 - 4x^2 = x^4 + 2x^2 + 9$

(4) $(x-y+z)(x+y-z) = (x-(y-z))(x+(y-z)) = x^2 - (y-z)^2 = x^2 - (y^2 - 2yz + z^2) = x^2 - y^2 + 2yz - z^2$

3. 最終的な答え

(1) $a^4 - 2a^2b^2 + b^4$

(2) $x^4 - 1$

(3) $x^4 + 2x^2 + 9$

(4) $x^2 - y^2 + 2yz - z^2$