与えられた6つの数式を展開せよ。

6問すべて解きます。

1. 問題の内容

与えられた6つの数式を展開せよ。

2. 解き方の手順

(1) $(x^2 + yz)(3yz + 2x^2)$

= $3x^2yz + 2x^4 + 3y^2z^2 + 2x^2yz$

= $2x^4 + 5x^2yz + 3y^2z^2$

(2) $(m^2 - 2m - 1)^2 (m^2-2m-1)$　この問題は (m²-2m-1)² ですね。

$(m^2 - 2m - 1)^2 = (m^2 - 2m - 1)(m^2 - 2m - 1)$

= $m^4 - 2m^3 - m^2 - 2m^3 + 4m^2 + 2m - m^2 + 2m + 1$

= $m^4 - 4m^3 + 2m^2 + 4m + 1$

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

= $[(x - y)(x + y)]^2(x^2 + y^2)^2$

= $(x^2 - y^2)^2(x^2 + y^2)^2$

= $[(x^2 - y^2)(x^2 + y^2)]^2$

= $(x^4 - y^4)^2$

= $x^8 - 2x^4y^4 + y^8$

(4) $(a + 3)(a - 2)(a^2 - a + 6)$

= $(a^2 + a - 6)(a^2 - a + 6)$

= $a^4 - a^3 + 6a^2 + a^3 - a^2 + 6a - 6a^2 + 6a - 36$

= $a^4 - a^2 + 12a - 36$

(5) $(4x - y + 1)(3x + y - 1)$

= $12x^2 + 4xy - 4x - 3xy - y^2 + y + 3x + y - 1$

= $12x^2 + xy - x - y^2 + 2y - 1$

(6) $(a + b - c - d)(a - b - c + d)$

= $[a - (c + d - b)][a - (c - d + b)]$

= $[a - (c + (d-b))][a - (c-(d-b))]$

= $[(a -c) - (d-b)][(a-c) + (d-b)]$

= $(a-c)^2 - (d-b)^2$

= $a^2 - 2ac + c^2 - (d^2 - 2bd + b^2)$

= $a^2 - b^2 + c^2 - d^2 - 2ac + 2bd$

3. 最終的な答え

(1) $2x^4 + 5x^2yz + 3y^2z^2$

(2) $m^4 - 4m^3 + 2m^2 + 4m + 1$

(3) $x^8 - 2x^4y^4 + y^8$

(4) $a^4 - a^2 + 12a - 36$

(5) $12x^2 + xy - x - y^2 + 2y - 1$

(6) $a^2 - b^2 + c^2 - d^2 - 2ac + 2bd$