Question

Three zeroes of a fifth degree polynomial function are one third, 4 minus 6 i, and negative 2 plus 11 i. Determine the remaining zeroes of the function.
(1 point)
Responses

4 + 6i, negative 2 minus 11 i
4 + 6 i, Image with alt text: negative 2 minus 11 i

negative one third, 4 + 6i, 2 + 11i
Image with alt text: negative one third , 4 + 6 i , 2 + 11 i

negative 4 plus 6 i, 2 minus 11 i
Image with alt text: negative 4 plus 6 i , Image with alt text: 2 minus 11 i

3, 4 + 6i, negative 2 minus 11 i

Answer 1

For polynomials with real coefficients, complex roots must occur in conjugate pairs. Given the zeroes of the polynomial:

1. \( \frac{1}{3} \) (real)
2. \( 4 - 6i \) (complex)
3. \( -2 + 11i \) (complex)

Since \( 4 - 6i \) is a root, its conjugate \( 4 + 6i \) must also be a root.

Similarly, for the root \( -2 + 11i \), its conjugate \( -2 - 11i \) must also be a root.

Thus, the remaining two zeroes are:
- \( 4 + 6i \)
- \( -2 - 11i \)

So the complete set of zeroes for this fifth degree polynomial is:
1. \( \frac{1}{3} \)
2. \( 4 - 6i \)
3. \( 4 + 6i \)
4. \( -2 + 11i \)
5. \( -2 - 11i \)

The correct response is:
**4 + 6i, -2 - 11i**.