Find the real roots

Question

^{x5 - 5ax + 4b = o. How many real roots does the equation has if a5 >b5 and a, b are real numbers.}

Answer 1

It would take time & a computer to determine all possibilities. Using some chosen numbers if a > b, a table of a few roots goes like this if a & b are shown in the columns and the result of roots is shown to the side.

a b Real Complex

2 1 3 2

3 1 3 2

3 2 3 2

4 1 3 2

4 2 3 2

4 3 3 2

10 1 3 2

10 9 3 2 This is not a total answer but the trend certainly indicates that given

equation and constraints, Real roots = 3 & Complex Roots = 2!