SOLUTION: Find a polynomial with integer coefficients that satisfies the given conditions. U has degree 5, zeros 1/2, −4, and −i, and leading coefficient 4; the zero −4 ha

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find a polynomial with integer coefficients that satisfies the given conditions. U has degree 5, zeros 1/2, −4, and −i, and leading coefficient 4; the zero −4 ha      Log On


   

Question 1099280: Find a polynomial with integer coefficients that satisfies the given conditions.
U has degree 5, zeros 1/2, −4, and −i, and leading coefficient 4; the zero −4 has multiplicity 2.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Each zero x%5Bk%5D means the factored form of the polynomial has
as many factors of the form %28x+-+x%5Bk%5D%29 as the multiplicity of the zero.
That means that the factored form of the polynomial includes
%28x-1%2F2%29%28x%2B4%29%5E2%28x%2BI%29
For each complex zero, the conjugate complex number is also a zero.
That means that i is also a zero, and that %28x-i%29 also appears as a factor.
So far we have the factors
%28x-1%2F2%29%28x%2B4%29%5E2%28x%2Bi%29%28x-i%29 ,
accounting for the degree 5,
with 1 for a leading coefficient.
We need to include 4 as a factor to have 4 for a leading coefficient.
The polynomial is

= .