Polynomails question from Cambridge (1 Viewer)

xMrRand0m · Apr 21, 2009

Hi, I need help with the following question:

The equation x^n + px - q = 0 has a double root. Show that (p/n) ^ n + (q/(n-1))^n-1 = 0

Thanks in advance! =)

Timothy.Siu · Apr 21, 2009

xMrRand0m said:
Hi, I need help with the following question:

The equation x^n + px - q = 0 has a double root. Show that (p/n) ^ n + (q/(n-1))^n-1 = 0

Thanks in advance! =)

f(x)=x^n+px-q
f'(x)=nx^(n-1)+p
=0 when x^(n-1)=-p/n
x=(-p/n)^(1/n-1)
(if a function has a double root, then that root is a root of the derivative)

i think thats right
and u just sub it into f(x)

jet · Apr 21, 2009

$x = \left(\frac{-p}{n} \right )^{n-1} \text{ is the root} \\ \text{Now, } x(x^{n-1} + p) = q \\ \left(\frac{-p}{n} \right )^{n-1} \left (p- \frac{p}{n} \right ) = q \\ \frac{p}{n}\left(\frac{-p}{n} \right )^{n-1} = \frac{q}{n-1} \\ \frac{p}{n}^{n-1}\left( \frac{p}{n}\right ) = -\left ( \frac{q}{n-1} \right )^{n-1} \\ \left ( \frac{p}{n} \right )^n = -\left (\frac{q}{n-1} \right ) \\ \left (\frac{p}{n} \right )^n + \left ( \frac{q}{n-1} \right)^{n-1} = 0$
Thats the rest.

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Polynomails question from Cambridge (1 Viewer)

xMrRand0m

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Timothy.Siu

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jet

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