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## Estimating roots

In lieu of a real blog post, which will have to wait for at least another two weeks, let me offer an estimation exercise: bound, as best you can, the unique positive real root of the polynomial

$\displaystyle x^{10000} + x^{100} - 1$.

The intermediate value theorem shows that $x \in (0, 1)$, which was the subject of a recent math.SE question that provided the inspiration for this question. I provide a stronger lower bound on $x$ using elementary inequalities and entirely by hand in an answer to the linked question, although I don’t try to improve the upper bound.