Question

Find a polynomial f(x) of degree 3 with real coefficients and the following zeros. -4, 1-i

Answer 1

the other root must be \(1+i\) since they come in conjugate pairs
now you have to find the quadratic that has \(1+i\) and \(1-i\) as a root
there are a couple of ways to do this
one is to work backwards and write
\[x=1+i\\
(x-1)^2=-1\\
x^2-2x+2=0\]

Answer 2

the other way is to memorize that if \(a+bi\) is a root of a quadratic, then the quadratic is
\[x^2-2ax+(a^2+b^2)\]

Answer 3

What do I with the -4?

Answer 4

Add another factor such that when x = -4, the function equals 0.