write a polynomial function of least degree with integral coefficients the zeros of which include 4 and 4-i

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