Find a third-degree polynomial equation with rational coefficients that has roots -4 and 3+i.please show work, i am so lost !!
Since this is a polynomial of degree three, by an application of the fundamental theorem of algebra we know that there are three roots; also all complex roots appear as conjugate pairs.
Thus the three roots are -4,`3+i,3-i`
If `k` is a root, then `(x-k)` is a factor by the factor theorem. Thus the factors of the polynomial are `(x+4),(x-3+i),(x-3-i)`
Then the function is :
`=a(x+4)(x^2-6x+9-i^2)` But `i^2=-1`
You have a choice of any `a` , letting `a=1` we have: