What two values to five significant figures should I use for a sign change test to see that $1.228$ (4sf) is a root of $x^4+x^3-2x^2+3x-5=0$?

Question

What two values to five significant figures should I use for a sign change test to see that $1.228$ (4sf) is a root of $x^4+x^3-2x^2+3x-5=0$?

$$f(x)=x^4+x^3-2x^2+3x-5$$ $$f(1.2279)=-0.207149...$$ $$f(1.2281)=-0.205145...$$ $$f(1.3)=0.5731$$ The root is somewhere in the region. After a search on WolframAlpha the root is approximately 1.2481. $$f(1.2480)=-1.4242...\times 10^{-3}$$ $$f(1.2482)=6.6726...\times 10^{-4}$$

I understand that the question may reveal a fault in the form however I am wondering whether there is something I am missing out on?

Answer 1

Without the computed value in hand the root could have been, say, $1.2283475...$ and it would have been $1.228$ to four significant digits yet outside your interval.

The correct interval specification is to include all the range where rounding to four significant digits would give $1.228$. That would be between $1.2275$ and $1.2285$. Put those bounds in and you should get 2/2 marks.