NapkinCalc

Precalculus B — Polynomials & Rational Functions

Polynomials — zeros & end behaviour

ELI5: a polynomial is a sum of power terms (like x³ − 4x). Two things describe it: its zeros (where it crosses the x-axis — at most "degree"-many of them) and its end behaviour (where the highest-power term drags it as x runs off to ±∞).

poly(x)=x3−4⋅xpoly\left(x\right) = x^{3} - 4 \cdot x degree 3 ⇒ up to three real zeros
zerosp=solve(x3−4⋅x,x)zeros_{p} = \mathrm{solve}\left(x^{3} - 4 \cdot x, x\right) = [0,2,−2][0, 2, -2] factor x(x−2)(x+2) ⇒ zeros {−2, 0, 2}
✓ pass max(zerosp)==2andmin(zerosp)==−2max(zeros_{p}) == 2 and min(zeros_{p}) == -2 the three crossings
poly(x)
-505-2-1012

x³ − 4x — three x-axis crossings, ends heading down-left / up-right

Real-world hook: polynomials fit curves to data and model smooth physical shapes — a roller-coaster hump, a beam's deflection, a trajectory with drag.