Let a and b be real numbers such that a not equal 0 Prove that not all the roots - Maths - Polynomials

Question

Let a and b be real numbers such that a not equal 0 Prove that not
all the roots of ax4 +bx3 + x2 + x + 1 = 0 can be real
Source:CRMO-2012

Pronay · Accepted Answer

Let $α_{1}$ , $α_{2}$ , $α_{3}$ and $α_{4}$ be the roots ${a x}^{4} + {b x}^{3} + x^{2} + x + 1 = 0$
Observe none of these is zero since their product is

$α_{1} α_{2} α_{3} α_{4} = \frac{1}{a}$

Then the roots of equation $x^{4} + x^{3} + x^{2} + b x + a = 0$ are

$β_{1} = \frac{1}{α_{1}}, β_{2} = \frac{1}{α_{2}}, β_{3} = \frac{1}{α_{3}}, β_{4} = \frac{1}{α_{4}}$

We have

$\sum_{j = 1}^{4} β_{j} = - 1$

$\sum_{1 \leq j < k \leq 4} β_{j} β_{k} = 1$

Hence

$\sum_{j = 1}^{4} β_{j}^{2} = {(\sum_{j = 1}^{4} β_{j})}^{2} - 2 \sum_{1 \leq j < k \leq 4} β_{j} β_{k} = 1 - 2 = - 1$

This shows that not all $β_{j}$ can be real. Hence not all $α_{j}$ 's can be real

Let a and b be real numbers such that a not equal 0. Prove that not all the roots of ax4 +bx3 + x2 + x + 1 = 0 can be real.Source:CRMO-2012