Question

There is a unique real root to the following. Solve with pen and paper.\[x^7+7x^3-7x^2+7x+1=0\]

Answer 1

Newton-Ralphson method..?

Or trial evaluation, eventually zooming in on a fair interval?

Answer 2

Either way, \(x\approx -0.125.\)

Answer 3

clearly no real solution for \(x\ge0\) and \(x\le −\frac{1}{7}\) so we must work on
\[−\frac{1}{7}<x<0\]
for this interval \(x^7≈0\) and also \(x^3≈0\)

solving \(−7x^2+7x+1=0\) gives \(x≈−0.126\)

Answer 4

actually from quadratic \[x=\frac{7-\sqrt{77}}{14}≈-0.126\]