Condition for real roots

**pankaj** · March 23rd 2009, 07:26 PM

Find the values of $a$ such that the equation

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

has $3$ different real roots.

**NonCommAlg** · March 24th 2009, 01:19 AM

Originally Posted by pankaj

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

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

**pankaj** · March 24th 2009, 06:54 AM

It means exactly 3 roots.Obviously one is a repeated root and the other two are different.Something like $x_{1},x_{1},x_{2},x_{3}$ .

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

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