find the polynomial with real coefficients of the smallest possible degree where i and 3 + i are zeros and the coefficient of the highest power is 1.

Question

anonymous · Answer

ok well if the zeros are complex then they must come with their conjugates.  zeros are therefore 
$$i, -i, 3+i,3-i$$

anonymous · Answer

in factored form you get
$$(x-i)(x+i)(x-(3+i))(x-(3-i))$$ and multiply out, which is not as bad as it looks

anonymous · Answer

first one is easy
$$(x+i)(x-i)=x^2+1$$

anonymous · Answer

second one is not too bad since you always have 
$$(x-(a+bi))(x-(a-bi))=x^2-2ax+(a^2+b^2)$$ so second one is 
$$x^2-6x+10$$

anonymous · Answer

and your last job is 
$$(x^2+1)(x^2-3x+10)$$ which i will let you do

anonymous · Answer

got it?

anonymous · Answer

got it !

anonymous · Answer

good.  i think i did  the other one too.  similar to this one

anonymous · Answer

okay , so ill have 3 seperate answers ? or just the last one..

anonymous · Answer

you have one answer but i did not do it.  you still have to multiply out 
$$(x^2+1)(x^2-6x+10)$$
"final answer" is $$x^4-6 x^3+11 x^2-6 x+10$$

anonymous · Answer

that is the polynomial you are looking for

anonymous · Answer

thats what i got , just checking. thanks a lot !