Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. -4, 2+(root 7) i know the answer is a trinomial because it makes sense 2+/- (root 7) but i need doing it .

x³ - 19x - 12

Step-by-step explanation:

complex roots occur as conjugate pairs

2 + is a root then 2 - is also a root

hence the factors of the polynomial are

(x + 4), (x - (2 + ) )(x - (2 - ))

and f(x) = (x + 4)(x - 2 - )(x - 2 + )

            = (x + 4)((x - 2)² - ()²)

            = (x + 4)(x² - 4x + 4 - 7)

            = (x + 4)(x² - 4x - 3)

            = x³ - 4x² - 3x + 4x² - 16x - 12 ← collect like terms

            = x³ - 19x - 12