What the app does
The domain is split at the roots found in the previous step, and one point inside each piece is tested. Because the function is continuous between consecutive roots, that single sample settles the sign for the whole interval.
Why it matters
This is the intermediate value theorem doing real work: it is what makes one test per interval enough, instead of checking infinitely many points. Knowing the sign tells you where the curve lies before you plot a single value.
Read further
- Intermediate value theorem Wikipedia
- Intermediate value theorem Wolfram MathWorld
- Polynomial inequalities Paul's Online Notes