Roots of polynomial function to 3rd power

**arze** · September 5th 2010, 09:20 PM

The three values of $\cosh x$ given by the equation
$a\cosh 3x+b\cosh 2x=0$
where a and b are non-zero, are $y_1$ , $y_2$ , and $y_3$ .
Show that $y_1y_2+y_2y_3+y_1y_3$ is independent of a and b.

$a\cosh 3x+b\cosh 2x=0$
$4a\cosh^3 x+2b\cosh^2 x-3a\cosh x-b=0$
I need to know the general formula for the roots? Or is there some simpler way?
Thanks!

**sa-ri-ga-ma** · September 5th 2010, 09:59 PM

In the cubic equation ax^3 + bx^2 + cx + d = 0, the sum of the product of roots αβ+βγ+γα = c/a.

Hence in the given problem y1y2 + y2y3 + y3y1 = -3a/4a = -3/4.

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