Find an nth degree polynomial with real coefficients satisfying the given conditions

Question

n=3; 4 and 5i are zeroes; f(2)= -174

Answer 1

The polynomial will be of the form asub3x^3 + asub2x^2 + asub1x + asub0. We'll leave off the a's for right now and construct the cubic polynomial (x-4)(x-5i)(x+5i) = (x-4)(x^2+25) = 0. Expanding this we get x^3 -4x^2 + 25 x -100 = 0. If we plug 2 into this polynomial, we get 8 - 16 + 50 - 100 = -58, which is 1/3 of -174. So our final polynomial is 3x^3 -12x^2 + 75x - 300.