Proof for no positive real roots

Question

Prove that $x^6$ + $x^4$ +$x^2$+ $x$ + $3$ has no positive real roots.

My attempt: I couldn't really think of a way to do this. I have looked at a similar problem but it had a cubic equation where you assign variables to the roots and use them to prove, but that method becomes tedious if you have to do it for this equation.

score 7 · Answer 1 · answered Mar 28 '17 at 06:51

7

If $x$ is positive then $x^2,x^4,x^6$ are also positive so $$x^6 + x^4 + x^2 + x + 3 \ge 3.$$

answered Mar 28 '17 at 06:51

user430043

81

Great answer!!!!!! – Amit Hegde Mar 28 '17 at 08:20

Proof for no positive real roots

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