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Answer:
third option
Step-by-step explanation:
Given f(x) has zeros x = a, x = b, then
the factors are (x - a), (x - b) and the polynomial is the product of the factors.
Complex zeros occur as conjugate pairs, thus
5 + 4i is a zero then 5 - 4i is a zero
The zeros are x = 3, x = - 13, x = 5 + 4i, x = 5 - 4i, thus the factors are
(x - 3), (x - (- 13)), (x - (5 + 4i)), (x - (5 - 4i)), that is
(x - 3), (x + 13), (x - 5 - 4i), (x - 5 + 4i)
= (x - 3), (x + 13), ((x - 5) - 4i), ((x - 5) + 4i)
The polynomial is the product of the factors
f(x) = (x - 3)(x + 13)((x - 5) - 4i)((x - 5) + 4i) ← expand factors in pairs
= (x² + 10x - 39)((x - 5)² - 16i²) → i² = - 1
= (x² + 10x - 39)(x² - 10x + 25 + 16)
= (x² + 10x - 39)(x² - 10x + 41) ← distribute
= - 10x³ + 41x² + 10x³ - 100x² + 410x - 39x² + 390x - 1599 ← collect terms
= - 98x² + 800x - 1599