To determine the number of real and imaginary zeros of the polynomial function f(x) = x³ + 5x² – 28x – 32, we can use the Rational Root Theorem and synthetic division, along with Descartes’ Rule of Signs.
First, we can apply Descartes’ Rule of Signs to find the possible number of positive and negative real zeros:
- Looking at f(x): The sign changes between +5x² and -28x, indicating there is one positive real zero.
- Now, we evaluate f(-x): f(-x) = -x³ + 5x² + 28x – 32. The sign changes occur between -x³ and +5x², and also between +28x and -32, which indicates there are two negative real zeros.
Using synthetic division, we can further analyze the function. Testing possible rational roots such as ±1, ±2, ±4, ±8, ±16, ±32 can help find the actual roots. After testing these, we find one real root, which tells us that there are two imaginary zeros remaining from the cubic function, since a cubic polynomial can have at most three roots.
In conclusion, the function f(x) = x³ + 5x² – 28x – 32 has:
- 1 real zero
- 2 imaginary zeros