The Anatomy of Higher Functions

Unit 7 reads polynomial and rational functions the way Unit 6 read parabolas — every behavior off the formula — and its one law is that factors carry all the information.

A polynomial's biography is written in two places. The leading term rules the far country: even degree matches the arms, odd degree opposes them, and the coefficient's sign points them. The factored form rules everything near: each zero's multiplicity decides crossing or touching, and degree n caps the turning points at n − 1 — the sketch of x³ − 3x + 2 needed exactly its allowed two. Division is the unit's engine for producing factors: long division ends in the Division Algorithm's receipt, synthetic division runs the same arithmetic in shorthand, and the Remainder and Factor Theorems fuse evaluation with division — f(1) = 0 and a clean division by x − 1 are one fact wearing two notations. Given one zero, the whole polynomial unravels a degree at a time until a quadratic remains, and quadratics always surrender. Rational functions then split the story of exclusion in two, and the split is the unit's finest distinction: a denominator zero whose factor survives simplification raises a wall — the vertical asymptote where outputs run to 51 and −49 within a tenth of the line — while a zero whose factor cancels leaves a hole, one missing point the simplified formula pretends to fill. The erased-hole error believes the pretense; the check remembers that the original divided zero by zero there. Far away, the degree comparison sets the horizontal ceiling. Variation closes the unit with the smallest models in the course — one constant k each, and the constant usually names something real: the 240 in a time-speed problem is the length of the road. Factors near, leading terms far, k in the middle of every proportion — the anatomy is complete.