Journal of Formalized Mathematics
Volume 15, 2003
University of Bialystok
Copyright (c) 2003 Association of Mizar Users

## Little Bezout Theorem (Factor Theorem)

Piotr Rudnicki

### Summary.

We present a formalization of the factor theorem for univariate polynomials, also called the (little) Bezout theorem: Let \$r\$ belong to a commutative ring \$L\$ and \$p(x)\$ be a polynomial over \$L\$. Then \$x-r\$ divides \$p(x)\$ iff \$p(r) = 0\$. We also prove some consequences of this theorem like that any non zero polynomial of degree \$n\$ over an algebraically closed integral domain has \$n\$ (non necessarily distinct) roots.

This work has been supported by NSERC Grant OGP9207.

#### MML Identifier: UPROOTS

The terminology and notation used in this paper have been introduced in the following articles [27] [37] [31] [8] [2] [26] [32] [15] [20] [38] [6] [7] [3] [9] [36] [33] [24] [23] [11] [21] [16] [19] [17] [18] [1] [12] [34] [28] [22] [10] [35] [4] [25] [39] [13] [29] [14] [30] [5]

#### Contents (PDF format)

1. Preliminaries
2. Canonical Ordering of a Finite Set
4. More on Polynomials
5. Little Bezout Theorem
6. Polynomials Defined by Roots

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