請教微積分多項式實數根的解法

1. 設f(x) = 6x^3 - 11x^2 - 19x - 6

因為f(3) = 6(3)^3 - 11(3)^2 - 19(3) - 6 = 0

根據因式定理，(x - 3)是f(x)的因子

6x^3 - 11x^2 - 19x - 6

= (x - 3)(6x^2 + 7x + 2)

= (x - 3)(2x + 1)(3x + 2)

所以，6x^3 - 11x^2 - 19x - 6 = 0

(x - 3)(2x + 1)(3x + 2) = 0

x = 3, -1/2 或 -2/3


2. sqrt[(x - 1)^2 + (4 - 0)^2] = 5

(x^2 - 2x + 1) + 16 = 25

x^2 - 2x - 8 = 0

(x - 4)(x + 2) = 0

x = 4 或 -2


3. y = x^2 + x - 2

y截距，代x = 0

y = -2

x截距，代y = 0

x^2 + x - 2 = 0

(x + 2)(x - 1) = 0

x = 1 或 -2