Repeated roots

**anthill** · October 26th 2010, 10:06 AM

Find all real values of k for which the equation $x^4-8x^3+kx^2-8x+1$ has at least one repeated root.

**Plato** · October 26th 2010, 10:37 AM

You do understand that this is not a homework service nor is it a tutorial service. PLease either post some of your own work on this problem or explain what you do not understand about the question.

**anthill** · October 27th 2010, 01:29 AM

First, $x^2[(x+\frac{1}{x})^2-8(x+\frac{1}{x})+(k-2)]=0$ where $x\neq0$

Then $x+\frac{1}{x} = 4 \pm \sqrt(18-k)$ and k= 14, -18 (rejected)

The thing I'm not sure about is whether there are any other real values of k apart from k=14 that would give at least one repeated root...

**HallsofIvy** · October 27th 2010, 04:34 AM

Originally Posted by anthill

First, $x^2[(x+\frac{1}{x})^2-8(x+\frac{1}{x})+(k-2)]=0$ where $x\neq0$

Then $x+\frac{1}{x} = 4 \pm \sqrt(18-k)$ and k= 14, -18 (rejected)

The thing I'm not sure about is whether there are any other real values of k apart from k=14 that would give at least one repeated root...

I'm not sure what you are trying to do here. You seem to be assuming that x must be an integer, which is not true. If x= a, say, is a repeated root then there is a factor of $(x- a)^2$ . Try writing $x^4- 8x^3+ kx^2- 8x+ 1= (x- a)^2(x^2+ cx+ d)$ .

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