If the given zeros are 4 and -2+√5, then the polynomial function of the least degree with integral coefficients that has these zeros can be obtained by using the fact that if a polynomial has a root at x=a, then it has a factor of (x-a).
Therefore, the polynomial function with these zeros can be written as:
(x - 4)(x - (-2+√5))(x - (-2-√5))
Expanding the factors using the difference of squares, we get:
(x - 4)((x + 2) - √5)((x + 2) + √5)
Simplifying the expression further, we get:
(x - 4)((x + 2)^2 - 5)
Expanding the square, we get:
(x - 4)(x^2 + 4x - 1)
Therefore, the polynomial function of the least degree with integral coefficients that has the zeros 4 and -2+√5 is:
f(x) = (x - 4)(x^2 + 4x - 1)
Expanding the factors, we get:
f(x) = x^3 + 4x^2 - x - 16