I recently stumbled across this great problem which is a must-try-to-solve problem if you want to practice your algebraic manipulation abilities. Αnd in case you are interested in advanced classical inequalities there are some useful identities it highlights such that However the subtle lie of the problem is that despite the numbers ,, , , and are all real, it can be shown that for the given values it is impossible for all , and to be real. Now I need to focus your attention specifically in those 3 real numbers: , , and . Do they remind you anything? No? Anything related with polynomials?
Well consider a third degree polynomial where , , are its roots (either real or not). Then you can rewrite the polynomial as . So , , and are the real coefficients of each term of a third degree polynomial!
I also though of Newton's Inequality (I'll try to explain it the best I can) which states that where represent the elementary symmetric polynomial divided by and is the number of variables. Therefore for we obtain where is a real number. Thus according to Newton's Inequality for it is
Now assume that has all its roots being real numbers. Then by substituting in we obtain
If is not satisfied we can conclude that some of the roots are not real numbers.
Taking the values of the problem we can create the polynomial . Let all its roots be real numbers, then by Newton's Inequality it should be Contradiction! Thus has two imaginary roots.
We can use this method of the ATTEMPT to determine the real roots of a polynomial, for all the polynomials of degree greater or equal to 3. For example for a fourth degree polynomial if either one OR more of the following inequalities do not hold then some of its roots are not real.
(Generally if is the degree of the given polynomial then you are able to check inequalities)
Caution! If the inequalities hold then we cannot conclude that the roots of a polynomial are real. My observation is that if at least one of the inequalities does not hold then the polynomial has imaginary roots :)