Question

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.

Answer 1

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

Answer 2

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

Answer 3

first one is easy
\[(x+i)(x-i)=x^2+1\]

Answer 4

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
\[x^2-6x+10\]

Answer 5

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

Answer 6

got it?

Answer 7

got it !

Answer 8

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

Answer 9

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

Answer 10

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 \[x^4-6 x^3+11 x^2-6 x+10\]

Answer 11

that is the polynomial you are looking for

Answer 12

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