Question

I'm having real trouble with these polynomial equations..
Z^3-5z^2+11z-15=0 and you know that it got a root that is z=1+2i

Answer 1

Remember: a polynomial has as many roots as its highest power. Here 3 is the highest power, so there are 3 roots

If a polynomial (with real coefficients) has a complex root a+bi, then it has another root a-bi (the complex conjugate)

so you know this polynomial has a root 1-2i

Answer 2

If you want to find the third root, you can do the following:
(x - (1+2i))(x - (1-2i)) will divide into the polynomial (if the roots are a and b , then (x-a)(x-b) divides into the original)

rewrite this as ((x-1) - 2i) ((x-1) + 2i)
multiply out to get (x-1)^2 + 4 = x^2 -2x +5

divide x^2 -2x +5 into x^3-5x^2+11x-15

(sorry about changing z to x, but it doesn't change things)