How do you find the complex zeros of a function?

Question

How do you find the complex zeros of a function after finding/knowing the positive and negative zeros?

Ex. f(x)=-3x^3+6x^2+8x-16

Answer 1

Once you get the function down to a quadratic, you can use the quadratic formula.

For f(x) = -3x³ + 6x² + 8x - 16 we can factor by grouping

f(x) = (-3x³ + 6x²) + (8x - 16)

f(x) = -3x²(x - 2) + 8(x - 2)

f(x) = (-3x² + 8)(x - 2)

To find the zeros, set the function equal to zero;

0 = (-3x² + 8)(x - 2)

Set each factor equal to zero

x - 2 = 0 means x = 2

-3x² + 8 = 0

3x² = 8

x² = 8/3

x = ± √(8/3) = ± √24/3 = ± 2√6/3

So this function has 3 real roots (no imaginary roots)