Question

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 help doing it thanks.

Answer 1

x³ - 19x - 12

Explanation:

complex roots occur as conjugate pairs

2 +
`√(7)` is a root then 2 -
`√(7)` is also a root

hence the factors of the polynomial are

(x + 4), (x - (2 +
`√(7)`) )(x - (2 -
`√(7)`))

and f(x) = (x + 4)(x - 2 -
`√(7)`)(x - 2 +
`√(7)`)

= (x + 4)((x - 2)² - (
`√(7)`)²)

= (x + 4)(x² - 4x + 4 - 7)

= (x + 4)(x² - 4x - 3)

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

= x³ - 19x - 12