Question

Find a polynomial of degree 2 with integer coefficients and zeros 5-i and 5+i.

Answer 1

you have lots of choices of methods here

Answer 2

one way is to multiply out
\[(x-(5+i))(x-(5-i))\] which is not as hard as it seems. but is kind of a pain

Answer 3

next choice is to work backwards
you know
\[x=5+i\]
\[x-5=i\]
\[(x-5)^2=i^2=-1\]
\[x^2-10x+25=-1\]
\[x^2-10x+26=0\] is the equation you started with

Answer 4

the other is to memorize that if \(a+bi\) is the zero of a polynomial of degree two with leading coefficient 1, that quadratic equation is
\[x^2-2ax+(a^2+b^2)=0\]

Answer 5

pick whichever method you like

Answer 6

Thank you for the very helpful and speedy response!

Answer 7

yw