LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-02

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A real polynomial vanishing at $a$ is divisible by $x - a$

Statement

If $p (x) = \sum_{k < n} a_{k} x^{k}$ and $p (a) = 0$ , then there is a real polynomial $q$ with $p (x) = (x - a) q (x)$ for every real $x$ . The conventions and prerequisite facts used below are recorded in Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials, Factorisation of $b^{n} - a^{n}$ , and the resulting Lipschitz estimate, Laws of finite sums and finite products.

Facts & Assumptions

Given: A polynomial $p$ and a real root $a$ .

Proof

technique · constructive

1.1

For each $k \geq 1$ , the power-difference factorization gives $x^{k} - a^{k} = (x - a) \sum_{j < k} a^{j} x^{k - 1 - j}$ .

given

1.2

Since $p (a) = 0$ , write $p (x) = \sum_{k < n} a_{k} (x^{k} - a^{k})$ .

algebra

2.1

Substitute the factorization from step 1.1 and collect the finite coefficient sums into a polynomial $q$ .

constructdischarge-construct∎

Depends on

Used by

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

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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

A real polynomial vanishing at aa is divisible by x−ax−a