SOLUTION: a)Find a 5th degreed polynomial with roots x=2, x=2+i and x=1-i. b)Now find the particular polynomial with the given roots in part a. that passes through (0,-60).

Algebra ->  Rational-functions -> SOLUTION: a)Find a 5th degreed polynomial with roots x=2, x=2+i and x=1-i. b)Now find the particular polynomial with the given roots in part a. that passes through (0,-60).      Log On


   

Question 729707: a)Find a 5th degreed polynomial with roots x=2, x=2+i and x=1-i.
b)Now find the particular polynomial with the given roots in part a. that passes through (0,-60).
Answer by josgarithmetic(39799) About Me  (Show Source):
You can put this solution on YOUR website!
The two complex roots with imaginary components require two additional complex roots, the conjugates.

Along with (x-2), (x-(2+i)) makes (x-(2-i)) also necessary.
(x-(1-i)) makes (x-(1+i)) also necessary.

The complex pairs can be turned into quadratic factors this way:
%28x-%282%2Bi%29%29+%2A+%28x-%282-i%29%29=x%5E2-4x%2B5;
%28x-%281-i%29%29+%2A+%28x-%281%2Bi%29%29=x%5E2-2x%2B2.

A function with the five expected roots may be
highlight%28p%28x%29=%28x-2%29%28x%5E2-4x%2B5%29%28x%5E2-2x%2B2%29%29.

Your next objective was to use that function as a starting relation to find a function to the degree 5 (?) which contains the point (0, -60). You may possibly need a constant factor with the polynomial to be able to have the function value equal to -60. Letting x=0 is simple; do this and then carry out the remaining multiplications.

-60=%280-2%29%280-0%2B5%29%280-0%2B2%29%2Ak, where k is some constant factor in case the original factors do not give -60.
By intent, I have not finished this work but it should be very easy.