✔ 最佳答案
Let f(x) = 6x³ + ax² + bx + 16
(x - 2) is a factor of f(x). Then, f(2) = 0
6(2)³ + a(2)² + b(2) + 16 = 0
2a + b = -32 …… [1]
(x + 2) is a factor of f(x). Then, f(-2) = 0
6(-2)³ + a(-2)² + b(-2) + 16 = 0
2a - b = 16 …… [2]
[1] + [2]:
4a = -16
a = -4
[1] - [2]:
2b = -48
b = -24
f(x) = 6x³ - 4x² - 24x + 16
= 6x³ + 12x² - 16x² - 32x + 8x + 16
= (6x³ + 12x²) - (16x² + 32x) + (8x + 16)
= 6x²(x + 2) - 16x(x + 2) + 8(x + 2)
= (x + 2)(6x² - 16x + 8)
= 2(x + 2)(3x² - 8x + 4)
= 2(x + 2)(x - 2)(3x - 2)
The answers:
a = -4 and b = -24
Remaining factors: 2 and (3x - 2)