Finding the root of a 5th degree polynomial

Question

I want to estimate the size for the root of the following polynomial: $$(x+0.893)^5+100(x+0.893)^3+10000(x+0.893)-9000.$$ I want to give an upper bound for its absolute value. I already know the root it "close" to $0$. My approach was to use Taylor series to aproximate functions of the form $(a+x)^n$ with $a^n+na^{n-1}x$. However, I'm afraid I might be finding an approximation of the root with smaller size than the real one, which I'm not interested in. Any ideas?

Answer 1

Would you not substitute $t=\frac{1}{10}(x+0.893)$? You would end up with:

$$t^5+t^3+t-0.09=0$$

i.e. $t(t^4+t^2+1)=0.09$. I believe this is enough of a hint that $t\approx 0.09$ and so $x\approx 0.007$.

Actually, $x=0.007$ will give you $t=0.09$ and so $t(1+t^2+t^4)\gt t=0.09$. On the other hand, $x=-0.007$ gives you $t=0.0886\lt 0.1$, so $t(1+t^2+t^4)\lt 0.886\cdot 1.0101\approx 0.895\lt 0.9$. Thus, the root must actually be between $-0.007$ and $0.007$.