SOLUTION: Write a polynomial function F of the least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. 1) {{{1- sqrt(5)}}} 2) {{{ sqrt (2)}}} , 3

Algebra ->  Test -> SOLUTION: Write a polynomial function F of the least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. 1) {{{1- sqrt(5)}}} 2) {{{ sqrt (2)}}} , 3      Log On


   

Question 825428: Write a polynomial function F of the least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.
1) 1-+sqrt%285%29
2) +sqrt+%282%29 , 3i
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
Initial impression here is that the irrational root would be treated similarly to the imaginary root. The help I give here is for question #1.

%281-sqrt%285%29%29%281%2Bsqrt%285%29%29=1%5E2-%28sqrt%285%29%29%5E2
1-5=-4

Hoping to use that,
F=%28x-%281-sqrt%285%29%29%29%28x-%281%2Bsqrt%285%29%29%29
%28x-1%2Bsqrt%285%29%29%28x-1-sqrt%285%29%29
%28x-1%29%5E2-%28sqrt%285%29%29%5E2
x%5E2-2x%2B1-5
x%5E2-2x-4

There it is!
highlight%28F=x%5E2-2x-4%29, and the TWO roots are 1-sqrt%285%29 and 1%2Bsqrt%285%29.