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Concept

Roots — where it appears

The values at which a polynomial takes the value zero.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

x²1.5x1.5x1.5²x + 1.5a squarex² + 3x + 1.5²= (x + 1.5)²x = 4, so the total is30.25

Completing the square, by completing a square

The step everybody is taught as an algebraic trick is a literal instruction about a literal square. There is a corner missing, its size is forced, and paying for it is the whole method.

algebra · completing the square radius 0.40 turns0 roots insideradius 10 turns0 roots insideradius 1.42 turns2 roots insideradius 23 turns3 roots insidep(z) = z³ − 2z + 2 — as the circle grows, the number of turns steps up from 0 to 3the count can only change by the image crossing the origin, and a crossing is a root

A loop that cannot miss the middle

Feed a circle into a polynomial and a closed loop comes out. A small circle gives a loop that does not enclose the origin; a large one gives a loop that goes round it as many times as the degree. Something has to happen in between, and that something is a root.

algebra · polynomial roots

Named alongside it

The objects these essays reach for when they reach for this one.

Algebra tilesAreaClosed curveCompleting the squareComplex numbersContinuityDegreeDissectionExistence proofFundamental theoremImaginary unitNonconstructive

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