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.

Question

(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
3, 4 + 6 i , Image with alt text: negative 2 minus 11 i

Answer 1

For a polynomial function with real coefficients, complex roots must occur in conjugate pairs. Given the zeros you have:

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

From the properties of polynomials with real coefficients, we also need the conjugates of the complex zeros:

• The conjugate of \( 4 - 6i \) is \( 4 + 6i \).
• The conjugate of \( -2 + 11i \) is \( -2 - 11i \).

Now we have four zeros:

1. \( \frac{1}{3} \)
2. \( 4 - 6i \)
3. \( -2 + 11i \)
4. \( 4 + 6i \) (conjugate of \( 4 - 6i \))
5. \( -2 - 11i \) (conjugate of \( -2 + 11i \))

However, since the polynomial is only of degree five, we can confirm that one of the roots is already known from the list provided. Therefore, we need to find out which is the remaining zero.

Given the provided zeroes, the remaining zeros must be:

• \( 4 + 6i \)
• \( -2 - 11i \)

Based on the choices you provided, the correct response for the remaining zeroes of the function is:

4 + 6i, -2 - 11 i