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Concept

Roots — where it appears

The values at which a polynomial takes the value zero. Their number is bounded by the degree, and over the complex numbers there are exactly that many when multiplicity is counted, which is the fundamental theorem of algebra.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Completing the square, as a square. An x by x square with the strip split in half and laid along two sides, leaving a square hole of side 1.5. Filling the hole costs 2.25 and buys a perfect square.

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 The image of four circles, turning 0 to 3 times. The polynomial applied to circles of four radii, each image drawn as a closed loop with the origin marked, and the number of times the loop goes round it.

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 3 roots, and the two numbers the coefficients already knew. The roots of a degree-3 polynomial, found numerically, with the point they average to. That average, and their product, are readable straight off the coefficients without finding the roots at all.

What the coefficients already know

Finding the roots of a polynomial is hard and often impossible in closed form. Reading off their sum, their product and how many of them are real is none of those things — those numbers are sitting in the coefficients, and no root-finding is required to get at them.

algebra · Polynomial roots

Named alongside it

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

Complex numbersDegreePolynomialAlgebra tilesAreaClosed curveCoefficientCompleting the squareContinuityDiscriminantDissectionExistence proof

All concepts