Find an equation for fx the polynomial of smallest degree wi

Find an equation for f(x), the polynomial of smallest degree with real coefficients such that f(x) bounces off of the x-axis at 5, breaks through the xx-axis at 2, has complex roots of 1+i and 4i and passes through the point (0,85)

Solution

Zeroes of f(x) are x = -85,-5,2,1+i,-4-i

Therefore, f(x) = (x+5)(x-2)(x+85)(x-1-i)(x+4+i)