TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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A nonzero real polynomial of degree $n$ has no more than $n$ distinct real roots

Statement

A nonzero real polynomial of degree $n$ has at most $n$ distinct real roots. The conventions and prerequisite facts used below are recorded in Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials, A real polynomial vanishing at $a$ is divisible by $x-a$ , The principle of mathematical induction.

Facts & Assumptions

Given: A nonzero real polynomial of degree $n$ .

Proof

technique · induction

1.1

At degree $0$ the polynomial is a nonzero constant and has no root.

base

1.2

Assume the claim at degree $n$ .

2.1

If a degree- $n+1$ polynomial has a root $a$ , the factor lemma writes it as $(x-a)q(x)$ with $q$ of degree $n$ ; every other root is a root of $q$ .

step 1.2given

3.1

The induction hypothesis gives at most $n$ other roots, hence at most $n+1$ roots in all.

discharge-induction∎

Depends on

Used by

For n≥1, 2¹⁻ⁿTₙ is the minimax monic polynomial of degree n on [-1,1] Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 37 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)

A nonzero real polynomial of degree nnn has no more than nnn distinct real roots