次の8つの式を展開せよ。 (1) $(x+2y+3)(x+2y-2)$ (2) $(2x+y-z)(2x-y+z)$ (3) $(a+2b)^2(a-2b)^2$ (4) $(3x-y)^2(3x+y)^2$ (5) $(a-b+c)^2$ (6) $(2x-y-2z)^2$ (7) $(x+1)(x+2)(x+3)(x+4)$ (8) $(a-b)(a+b)(a^2+b^2)$

1. 問題の内容

次の8つの式を展開せよ。

(1) $(x+2y+3)(x+2y-2)$

(2) $(2x+y-z)(2x-y+z)$

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

(4) $(3x-y)^2(3x+y)^2$

(5) $(a-b+c)^2$

(6) $(2x-y-2z)^2$

(7) $(x+1)(x+2)(x+3)(x+4)$

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

2. 解き方の手順

(1) $x+2y=A$ とおくと、

$(x+2y+3)(x+2y-2) = (A+3)(A-2) = A^2 + A - 6 = (x+2y)^2 + (x+2y) - 6 = x^2 + 4xy + 4y^2 + x + 2y - 6$

(2) $2x=A, y-z=B$ とおくと、

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

(3) $(a+2b)^2(a-2b)^2 = [(a+2b)(a-2b)]^2 = (a^2 - 4b^2)^2 = a^4 - 8a^2b^2 + 16b^4$

(4) $(3x-y)^2(3x+y)^2 = [(3x-y)(3x+y)]^2 = (9x^2 - y^2)^2 = 81x^4 - 18x^2y^2 + y^4$

(5) $(a-b+c)^2 = (a-b+c)(a-b+c) = a(a-b+c) -b(a-b+c) + c(a-b+c) = a^2 -ab + ac -ba + b^2 -bc + ca - cb + c^2 = a^2 + b^2 + c^2 - 2ab - 2bc + 2ca$

(6) $(2x-y-2z)^2 = (2x-y-2z)(2x-y-2z) = 2x(2x-y-2z) -y(2x-y-2z) -2z(2x-y-2z) = 4x^2 - 2xy - 4xz - 2xy + y^2 + 2yz - 4xz + 2yz + 4z^2 = 4x^2 + y^2 + 4z^2 - 4xy - 8xz + 4yz$

(7) $(x+1)(x+2)(x+3)(x+4) = [(x+1)(x+4)][(x+2)(x+3)] = (x^2 + 5x + 4)(x^2 + 5x + 6)$

$x^2 + 5x = A$ とおくと、

$(A+4)(A+6) = A^2 + 10A + 24 = (x^2+5x)^2 + 10(x^2+5x) + 24 = x^4 + 10x^3 + 25x^2 + 10x^2 + 50x + 24 = x^4 + 10x^3 + 35x^2 + 50x + 24$

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

3. 最終的な答え

(1) $x^2 + 4xy + 4y^2 + x + 2y - 6$

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

(3) $a^4 - 8a^2b^2 + 16b^4$

(4) $81x^4 - 18x^2y^2 + y^4$

(5) $a^2 + b^2 + c^2 - 2ab - 2bc + 2ca$

(6) $4x^2 + y^2 + 4z^2 - 4xy - 8xz + 4yz$

(7) $x^4 + 10x^3 + 35x^2 + 50x + 24$

(8) $a^4 - b^4$