Indentities一問

(ax + 2)(x^2 + bx + c) ≡ ax^3 + dx^2 + e
(ax + 2)(x^2 + bx + c)
= ax^3 + abx^2 + acx + 2x^2 + 2bx + 2c
= ax^3 + (ab + 2)x^2 + (ac + 2b)x + 2c ≡ ax^3 + dx^2 + e
ab + 2 = d ... (1)
ac + 2b = 0 ... (2)
2c = e ... (3)
when c = 1, e = 2,
(2) => a + 2b = 0 => a = -2b
(1) => (-2b)(b) + 2 = d
d = 2 - 2b^2
when b = 1, a = -2, d = 0
(-2x + 2)(x^2 + x + 1) ≡ -2x^3 + 0x^2 + 2
when c = -1, e = -2
(2) => -a + 2b = 0 => a = 2b
(1) => (2b)(b) + 2 = d
d = 2 + 2b^2
when b = 2, a = 4, d = 10
(4x + 2)(x^2 + 2x - 1) ≡ 4x^3 + 10x^2 - 2

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