Final answer:
To find the integer roots of the given function, use the Integer Root Theorem to find the possible rational roots. Test each of these values in the function to identify the integer roots. To find the complex roots, divide the function by the quadratic factor and solve the resulting equation. Therefore, the complex roots of the function are -10, 4, 0, and 2.
Step-by-step explanation:
To find the integer roots of the given function, we can use the Integer Root Theorem. According to the theorem, any rational root of a polynomial equation with integer coefficients will be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In this case, the constant term is -80 and the leading coefficient is 1. The factors of -80 are ±1, ±2, ±4, ±5, ±8, ±10, ±16, ±20, ±40, ±80, and the factors of 1 are ±1. Therefore, the possible rational roots of the function are:
- p/q = ±1/±1 = ±1
- p/q = ±2/±1 = ±2
- p/q = ±4/±1 = ±4
- p/q = ±5/±1 = ±5
- p/q = ±8/±1 = ±8
- p/q = ±10/±1 = ±10
- p/q = ±16/±1 = ±16
- p/q = ±20/±1 = ±20
- p/q = ±40/±1 = ±40
- p/q = ±80/±1 = ±80
Now, we substitute each of these rational values into the function to see if they are roots. By testing each value, we find that the integer roots of the function are -10 and 4.
To find the complex roots of the function, one approach is to use polynomial division to divide the given function by the quadratic factor (x - r), where r is an integer root.
By synthetic division with -10, we divide the function f(x) = x² + 5x³ - 12x² - 76x - 80 by (x + 10).
The resulting quotient is x² - 5x² + 4x - 8. To determine the remaining complex roots, we solve the equation x² - 5x + 4x - 8 = 0 by factoring.
We can factor this equation as (x² - 5x) + (4x - 8) = 0, which simplifies to x(x - 5) + 4(x - 2) = 0. Setting each factor equal to zero, we find two additional roots of the function: x = 0 and x = 2. Therefore, the complex roots of the function are -10, 4, 0, and 2.