Math
posted by Anon on .
Create a 3rd degree polynomial with real coefficients that has roots 1 and 4i. Write your answer in form ax^3 + bx^2 + cx+ d .

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