Question

What are the real and complex zeros of the function y = x³ - 3x² + x - 3?

1) -3, -1, and 1
2) -3, i, and -i
3) 3, i, and -i
4) -3, -2, and 3

Answer 1

Both the answer and how to get the answer are in the image above

Answer 2

The real zeros of the function are -3 and 1. The complex zeros are 3 + i and 3 - i.

Step-by-step explanation:

The given function is y = x³ - 3x² + x - 3. To find the real and complex zeros of the function, we set y equal to zero and solve for x.

0 = x³ - 3x² + x - 3

Using synthetic division or the rational root theorem, we can find that the real zeros are x = -3, x = 1.

To find the complex zeros, we can use the quadratic formula to solve for the imaginary parts. The discriminant is given by b² - 4ac, where a = 1, b = -3, and c = -3. The discriminant is positive, which means there are two complex zeros.

Using the quadratic formula, we find that the complex zeros are x = (3 ± i).