Friday, October 11, 2019

algebra precalculus - Sum of roots rational but product irrational



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 x48x3+18x28x7 has roots x1=12,x2=3+2,x3=1+2,x4=32.
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 uQ, 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)=(x2ux+a)(x2vx+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)vbv+bu, we see that b(uv) 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)=x4ux3+ax2) as aQ, and if x4=0 and x30, we get that x3Q, and x3Q is not possible again, as P(x)=(xx3)(x2ux+a)=x+..x3ax.


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