How to find polynomial with given zeroes

Question

I'm given the zeroes -1 and 2-5i and I need to find the polynomial. I'm given part of the polynomial

(x^3-3x^2+__x+29), and I cannot find out how to do it, even with looking up tutorials. What do I do?

Answer 1

x=-1 is a solution so x+1 is a factor

since 2-5i is a solution, ASSUMING the polynomial

has REAL coefficients, then 2 + 5i is also a solution...

The polynomial is f(x) = (x+1) [ (x-2)+ 5i ][ (x-2) - 5i ]

= (x+1) [ (x-2)^2 + 25 ]

= (x+1) [ x^2 - 4x + 4 + 25]

= (x+1) (x^2 -4x + 29)

x^3 + x^2

- 4x^2 - 4x

+ 29x + 29

x^3 - 3x^2 + 25x + 29