Final Answer:
The roots of the function g(x) = x³ - 12x² + 21x + 98 are x = -2, x = 7, and x = -8.
Step-by-step explanation:
Since we know that g(-2) = 0, this means that x = -2 is a root of the function. We can use this information to factor the polynomial:
g(x) = (x - (-2))(x² + ax + b)
Expanding the product, we get:
g(x) = x³ + (a + 2)x² + (-2a - 2b)x + 2b
Comparing this to the given form of g(x), we can equate the coefficients of like terms:
a + 2 = -12
-2a - 2b = 21
2b = 98
Solving for a and b, we get:
a = -14
b = 49
Substituting these values back into the factored form, we get:
g(x) = (x - (-2))(x² - 14x + 49)
Further factoring the quadratic, we get:
g(x) = (x + 2)(x - 7)
Therefore, the roots of the function g(x) are x = -2, x = 7, and x = -8.