Hey there! In this problem, the solutions, both real and imaginary are provided. We have to work backwards to create a polynomial function which matches these solutions:
f(x) = (x - 6), (x + [5 + 2i]), (x + [5 - 2i])
Create a difference of squares,
f(x) = (x - 6), ([x + 5] + 2i), ([x + 5] - 2i)
f(x) = (x - 6), ([x + 5]^2 + 4)
[(x + 5)(x + 5)] + 4
x^2 + 5x + 5x + 25 + 4
x^2 + 10x + 29
f(x) = (x-6), (x^2 + 10x + 29)
f(x) = x^3 + 10x^2 + 29x - 6x^2 - 60x - 174
Combine like terms,
f(x) = x^3 + 4x^2 + 29x - 174
There we have our answer, a polynomial of degree 3.
[tex]f(x) = x^3 + 4x^2 + 29x - 174[/tex]
