与えられた4つの多項式の積を展開し、空欄を埋める問題です。 (1) $(x^2 + 2x - 6)(3x - 1)$ (2) $(2x^2 - 5x + 1)(x^2 - x - 3)$ (3) $(x^2 - 4x + 2)(x^2 + 4x + 2)$ (4) $(x + 1)(x - 1)(x - 2)(x - 4)$

1. 問題の内容

与えられた4つの多項式の積を展開し、空欄を埋める問題です。

(1) $(x^2 + 2x - 6)(3x - 1)$

(2) $(2x^2 - 5x + 1)(x^2 - x - 3)$

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

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

2. 解き方の手順

(1) $(x^2 + 2x - 6)(3x - 1) = 3x^3 - x^2 + 6x^2 - 2x - 18x + 6 = 3x^3 + 5x^2 - 20x + 6$

(2) $(2x^2 - 5x + 1)(x^2 - x - 3) = 2x^4 - 2x^3 - 6x^2 - 5x^3 + 5x^2 + 15x + x^2 - x - 3 = 2x^4 - 7x^3 + 0x^2 + 14x - 3$

(3) $(x^2 - 4x + 2)(x^2 + 4x + 2) = (x^2 + 2 - 4x)(x^2 + 2 + 4x) = (x^2 + 2)^2 - (4x)^2 = x^4 + 4x^2 + 4 - 16x^2 = x^4 - 12x^2 + 4$

(4) $(x + 1)(x - 1)(x - 2)(x - 4) = (x^2 - 1)(x^2 - 6x + 8) = x^4 - 6x^3 + 8x^2 - x^2 + 6x - 8 = x^4 - 6x^3 + 7x^2 + 6x - 8$

3. 最終的な答え

(1) 1: 3, 2: 5, 3: 2, 4: 0, 5: 6

(2) 6: 2, 7: 7, 8: 0, 9: 14, 10: 3

(3) 11: -, 12: 12, 13: 4

(4) 14: -6, 15: 7, 16: 6, 17: -8