# Thread: Problem Involving Polynomials and Their Roots

1. ## Problem Involving Polynomials and Their Roots

If I need to find the a polynomial with real coefficients that possesses the roots -2 and 2-3i, I assume that I must do the following ...

H(x) = (x + 2) (x - (2 - 3i) (x - (2 + 3i)
= (x + 2) ((x - 2) - 3i) ((x - 2)+ 3i)
= (x + 2) ((x - 2)^2 - (3i)^2)
= (x + 2) ((x^2 - 4x + 4) - 9i^2)
= (x + 2) (x^2 - 4x + 4 - 9(-1))
= (x + 2) (x^2 - 4x +13)
= x^3 - 2x^2 + 5x + 26

Is this correct or am I missing something? Any suggestions or advice would be greatly appreciated. Thanks in advance.

2. Originally Posted by headless chickens
If I need to find the a polynomial with real coefficients that possesses the roots -2 and 2-3i, I assume that I must do the following ...

H(x) = (x + 2) (x - (2 - 3i) (x - (2 + 3i)
= (x + 2) ((x - 2) - 3i) ((x - 2)+ 3i)
= (x + 2) ((x - 2)^2 - (3i)^2)
= (x + 2) ((x^2 - 4x + 4) - 9i^2)
= (x + 2) (x^2 - 4x + 4 - 9(-1))
= (x + 2) (x^2 - 4x +13)
= x^3 - 2x^2 + 5x + 26

Is this correct or am I missing something? Any suggestions or advice would be greatly appreciated. Thanks in advance.
This is correct, it is a polynomial with real coefficients with the given roots, any other polynomial with those roots and real coefficients will be of the form of a polynomial with real coefficients times this one.

CB