2023/019) Let a,b,c,d be any four real numbers but not all equal to zero. Prove that the roots of the polynomial $x^6 + a x^ 3 + bx^2 + c x + d = 0$ cannot all be real.

Because it is a degree 6 polynomial so  there are six roots let them be $x_1,x_2,x_3,x_4,x_5,x_6$ 

Because coefficient of  $x^5$  is zero so sum of roots is zero

So $\sum_{i=1}^{6}x_i = 0\cdots(1)$

Because coefficient of  $x^45$  is zero so double sum of roots is zero

So $\sum_{i=1}^{6}\sum_{j=1. j\ne i}^{6}x_i x_j = 0\cdots(2)$

Hence from (1) and (2) we have $\sum_{i=1}^{6}x_i^2 = 0$

If roots are real  all $x_i$ are zero which is impossible if a,b,c,d are non zero

Hence there is a contradiction and all roots cannot be real