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

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- Oct 26th 2010, 10:06 AManthillRepeated roots
Find all real values of k for which the equation $\displaystyle x^4-8x^3+kx^2-8x+1$ has at least one repeated root.

- Oct 26th 2010, 10:37 AMPlato
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.

- Oct 27th 2010, 01:29 AManthill
First, $\displaystyle x^2[(x+\frac{1}{x})^2-8(x+\frac{1}{x})+(k-2)]=0$ where $\displaystyle x\neq0$

Then $\displaystyle x+\frac{1}{x} = 4 \pm \sqrt(18-k)$ 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... - Oct 27th 2010, 04:34 AMHallsofIvy
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 $\displaystyle (x- a)^2$. Try writing $\displaystyle x^4- 8x^3+ kx^2- 8x+ 1= (x- a)^2(x^2+ cx+ d)$.