# Condition for real roots

• Mar 23rd 2009, 07:26 PM
pankaj
Condition for real roots
Find the values of $\displaystyle a$ such that the equation

$\displaystyle x^4+ax^3+bx^2+ax+1=0$

has $\displaystyle 3$ different real roots.
• Mar 24th 2009, 01:19 AM
NonCommAlg
Quote:

Originally Posted by pankaj

Find the values of $\displaystyle a$ such that the equation $\displaystyle x^4+ax^3+bx^2+ax+1=0$ has $\displaystyle 3$ different real roots.

did you mean "exactly" or "at least" 3 distinct real roots? anyway, the main idea is to note that if $\displaystyle x$ is a root, then $\displaystyle \frac{1}{x}$ will be a root too.

[for those who probably don't know, polynomials of this form can be easily reduced to a quadratic equation by putting $\displaystyle x+\frac{1}{x}=t.$]
• Mar 24th 2009, 06:54 AM
pankaj
It means exactly 3 roots.Obviously one is a repeated root and the other two are different.Something like $\displaystyle x_{1},x_{1},x_{2},x_{3}$.

$\displaystyle x_{1},x_{2},x_{3}$ being all distinct