Answer:
x³ - 12x² + 49x - 68
Step-by-step explanation:
Complex roots (zeros) occur in conjugate pairs
Thus 4 - i is a zero then 4 + i is a zero
Given the zeros are x = 4, x = 4 ± i, then
the factors are (x - 4), (x - (4 - i)) and (x - (4 + i))
The polynomial is the product of the factors, so
p(x) = (x - 4)(x - 4 + i)(x - 4 - i) ← expand the second pair of factors
= (x - 4)((x - 4)² - i²) → note i² = - 1
= (x - 4)(x² - 8x + 16 + 1)
= (x - 4)(x² - 8x + 17) ← distribute
= x³ - 8x² + 17x - 4x² + 32x - 68 ← collect like terms
p(x) = x³ - 12x² + 49x - 68
