Posted by Anon on Friday, May 24, 2013 at 5:49am.
The general equation for the function is:
f( x ) = a ( x - x1 ) ( x - x2 )( x - x3 )
where x1, x2 and x3 are the roots and a is the coefficient of x ^ 3
There is a theorem in algebra called the Imaginary Zeros Theorem.
It says that for a polynomial that is restricted to real coefficients then for any imaginary zero (root) that exists then its conjugate also exists.
This means that the roots you gave :
-1 and ( 4 + i ), also includes the congugate root:
( 4 - i )
This is the conjugate to ( 4 + i )
Since there are no other restrictions to the polynomial you can construct the roots in factored form like this:
f ( x ) = a [ x - ( - 1 ) ] ( x - 4 i ) ( x + 4 i ) =
a ( x + 1 ) ( x - 4 i ) ( x + 4 i ) =
a ( x + 1 ) ( x * x - x * 4 i + 4 i * x + 4 i * ( - 4 i ) =
a ( x + 1 ) ( x ^ 2 - 4 i x + 4 i x - 16 i ^ 2 ) =
a ( x + 1 ) [ x ^ 2 - 4 i x + 4 i x - 16 * ( - 1 ) ] =
a ( x + 1 ) ( x ^ 2 + 16 ) =
a ( x ^ 2 * x + x ^ 2 * 1 + 16 * x + 16 * 1 ) =
a ( x ^ 3 + x ^ 2 + 16 x + 16 )
a can be any real number.
If a = 1 then
f ( x ) = x ^ 3 + x ^ 2 + 16 x + 16