Which answer best describes the complex zeros of the polynomial function? f(x) = x³ - x² + 6x - 6
The function has one real zero and two nonreal zeros. The graph of the function intersects the x-axis at exactly two locations.
The function has three real zeros. The graph of the function intersects the x-axis at exactly three locations.
The function has two real zeros and one nonreal zero. The graph of the function intersects the x-axis at exactly one location.
The function has one real zero and two nonreal zeros. The graph of the function intersects the x-axis at exactly one location.

Solution:

Given function f(x) = x³ - x² + 6x - 6

Put x = 1; f(1) = 1³ - 1² + 6(1) - 6 = 0

x = 1 is real zeros of given polynomial.

To find remaining roots we can use synthetic division.

1 │ 1 -1  6  -6
   │ 0  1  0  6
-------------------
   │ 1  0  6  0

Now, quotient is x² + 6

Take x² + 6 = 0

x² = -6

x = ± √(-6) = ±√6i

⇒ The other two roots are imaginary.

Therefore, the function has one real zero and two non-real zeros.

The graph of the function intersects the x-axis at exactly one location.

Which answer best describes the complex zeros of the polynomial function? f(x) = x³ - x² + 6x - 6

Summary:

“The function has one real zero and two nonreal zeros. The graph of the function intersects the x-axis at exactly one location.”→ best describes the complex zeros of the polynomial function f(x) = x³ - x² + 6x - 6