Question

The polynomial function y = x^3 -3x^2 + 16x - 48 has only one non-repeated x-intercept. What do you know about the complex zeros of the function?

Answer 1

In order to find the zeros of the function, we can factor the polynomial by grouping.

x³ - 3x² + 16x - 48

= x²(x - 3) + 16(x - 3)

= (x² + 16)(x - 3)

Solve for the zeros:

1) x - 3 = 0

Solution: x = 3

The non repeated x-intercept is 3. The polynomial has the zero of 3 multiplicity 1.

2) x² + 16 = 0

x² = -16

x = ±√-16

x = ± i√16

x = ± 4i

Solution: x = 0 + 4i and and x = 0 - 4i

The complex zeros of the function in a + bi form is 0 ± 4i. The function has a total of two complex zeros.