To determine the nonreal roots of the polynomial [tex]\(3x^4 - 6x^3 + 3x^2 - 54x - 216 = 0\)[/tex] given that its real roots are -2 and 4, we can proceed with the following steps:
1. Recognize that the polynomial can be factored using its real roots. Thus, the polynomial can be expressed as follows:
[tex]\[
(x + 2)(x - 4)(\text{quadratic polynomial with nonreal roots})
\][/tex]
2. Assume the quadratic factor has nonreal roots. Let the quadratic polynomial have the form:
[tex]\[
ax^2 + bx + c
\][/tex]
3. Multiply the factors:
[tex]\[
3(x + 2)(x - 4)(ax^2 + bx + c)
\][/tex]
4. To find the quadratic polynomial, we compare the product coefficients with those of the original polynomial [tex]\(3x^4 - 6x^3 + 3x^2 - 54x - 216\)[/tex].
5. The quadratic polynomial will yield complex conjugate pairs since the coefficients of the polynomial are real numbers.
6. The nonreal roots of this polynomial turn out to be:
[tex]\[
-3i \quad \text{and} \quad 3i
\][/tex]
Thus, the nonreal roots of the given polynomial [tex]\(3x^4 - 6x^3 + 3x^2 - 54x - 216 = 0\)[/tex] are:
[tex]\[
(-2.7755575615628914e-17 + 2.9999999999999987i) \quad \text{and} \quad (-2.7755575615628914e-17 - 2.9999999999999987i)
\][/tex]
which are approximately:
[tex]\[
3i \quad \text{and} \quad -3i
\][/tex]
Therefore, the correct nonreal roots are:
[tex]\[
-3i \quad \text{and} \quad 3i
\][/tex]
Hence, the correct answer is:
[tex]\[
-3i, 3i
\][/tex]
