Form a third-degree polynomial function with real coefficients such that 3+i and 6 are zeros?

Question

mathmale · Answer

Hint:  If 3+i is a root, then 3-i is also a root.  These are "conjugate pairs."
If 6 is a zero, how would you write a factor of this polynomial that includes 6?  That includes 3+i?   That includes 3-i?

Please show your work.

hockeychick23 · Answer

(x-6)[(x-3)-i][(x-3)-i] ?

mathmale · Answer

Looks good.  If mult. together, will these terms produce a 3rd order poly?  Try it.

hockeychick23 · Answer

x^3-12x^2-2x^2i+18xi+44x-48-36i ?

mathmale · Answer

You'll need to combine like terms.  x^3 stands by itself, but there are two terms in x^2.  If you do this problem properly, ' i ' will likely disappear from your expression completely.  Work at it, please.

hockeychick23 · Answer

(x-6)[(x-3)-i][(x-3)-i] 
(x-6)[(x-3)^2-2]
(x-6)(x^2-6x+9)-2
x^3-6x^2+9x-6x^2+36x^2-54-2
x^3+24x^2+9x-56

@mathmale can you tell me if i did this right please?