Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the given zeros for 3, 5 + square root of 5

The general form of a polynomial is: x² + bx + c = 0. Substituting the roots,

3² + b (3) + c = 0

3b + c = - 9 - - > eqn 1

(5 + √5) ² + (5 + √5) (b) + c = 0

(5+√5) b + c = - 30-10√5 - - > eqn 2

Solving both equations simultaneously by subtracting eqn 2 from eqn 1,

(-2 - √5) b = 21 + 10√5

b = - 8 - √5

Using eqn 1,

3 (-8 - √5) + c = - 9

-24 - 3√5 + c = - 9

c = - 9 + 24 + 3√5

c = 15 + 3√5

Hence, the polynomial is: f (x) = x² + (-8 - √5) x + (15 + 3√5).