Suppose that x1,x2,x3,x4 are the real roots of a polynomial with integer coefficients of degree 4, and x1+x2 is rational while x1x2 is irrational. Is it necessary that x1+x2=x3+x4?
For example, the polynomial x4−8x3+18x2−8x−7 has roots x1=1−√2,x2=3+√2,x3=1+√2,x4=3−√2.
It holds that x1+x2 is rational while x1x2 is irrational, and we have x1+x2=x3+x4.
Answer
Suppose that all the roots are non zero. Put x1+x2=u, a=x1x2, x3+x4=v, b=x3x4, we suppose that u∈Q, then it is also the case for x3+x4, as the sum of all the roots is rational, and that a,b are irrationals (as the product ab is rational, if a is irrational, then it is also the case for b). The polynomial with roots x1,x2,x3,x4 is
P(x)=(x2−ux+a)(x2−vx+b)=x4−(u+v)x3+(a+uv+b)x2−(av+bu)x+ab
By hypothesis, we get that a+b+uv, av+bu, and ab are in Q. Writing
av+bu=(a+b)v−bv+bu, we see that b(u−v) is rational. As b is not, this imply u=v.
If now x3x4=0, then it is not possible that x3=x4=0,(in this case P(x)=x4−ux3+ax2) as a∉Q, and if x4=0 and x3≠0, we get that x3∈Q, and x3∈Q is not possible again, as P(x)=(x−x3)(x2−ux+a)=x⁴+..−x3ax.
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