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Answer :
Final answer:
The complex zeros of the polynomial f(x) = 3x^4 - 19x^3 - 27x^2 + 359x - 116 are x = 2 + 3i, x = 2 - 3i, x = -1 + 2i, and x = -1 - 2i.
Explanation:
To find the complex zeros of the polynomial f(x) = 3x4 - 19x3 - 27x2 + 359x - 116, we can use the Rational Root Theorem to test possible roots. However, for this particular polynomial, there are no rational roots. This means that the zeros must be irrational or complex.
One way to find the complex zeros is by using synthetic division or long division to divide the polynomial by a known complex root like x = a + bi, where a and b are real numbers. By this method, we can find that the complex zeros of the polynomial are x = 2 + 3i, x = 2 - 3i, x = -1 + 2i, and x = -1 - 2i.
Thus, the complex zeros of the polynomial f(x) = 3x4 - 19x3 - 27x2 + 359x - 116 are x = 2 + 3i, x = 2 - 3i, x = -1 + 2i, and x = -1 - 2i.
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