construct a 3rd degree polynomial with zeros of 4-i, 4+i, 2 and a leading coefficent of -4

Question

I need to construct a 3rd degree polynomial with zeros of 4-i, 4+i, 2 and a leading coefficent of -4, I have no ideA HOW to do this

Answer 1

There are three zeroes and it's a third degree polynomial so there are no more zeroes. The complex zeroes can be multiplied together, cancelling out the complex terms:

(x-4+i)(x-4-i)=(x-4)^2+1=x^2-8x+17.

Now multiply by the final factor (x-2) to give x^3-8x^2+17x-2x^2+16x-34=x^3-10x^2+33x-34. The leading coefficient is -4 so we need to multiply through by this number: -4x^3+40x^2-132x+136.