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HomeHomework HelpmathematicsUnique Factorization of Polynomials

Unique Factorization of Polynomials

Unique factorization of polynomials refers to the property that every polynomial can be represented uniquely as a product of irreducible polynomials over a given field, up to the order of the factors and multiplication by non-zero scalars.

intermediate
2 hours
Mathematics
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Overview

Unique factorization of polynomials is a fundamental concept in algebra that allows us to express polynomials as products of irreducible polynomials. This concept is similar to prime factorization in integers, where every integer can be uniquely expressed as a product of prime numbers. Understanding...

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Key Terms

Polynomial
An expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.

Example: 3x² + 2x - 5

Irreducible Polynomial
A polynomial that cannot be factored into simpler polynomials over its coefficient field.

Example: x² + 1 is irreducible over the reals.

Degree of a Polynomial
The highest power of the variable in a polynomial.

Example: In 4x³ + 2x² - x, the degree is 3.

Factor Theorem
A theorem stating that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0.

Example: For f(x) = x² - 4, (x - 2) is a factor since f(2) = 0.

Polynomial Division
The process of dividing one polynomial by another, similar to long division with numbers.

Example: Dividing x² + 5x + 6 by x + 2.

Synthetic Division
A simplified form of polynomial division that is used when dividing by linear factors.

Example: Using synthetic division to divide x³ - 3x² + 4 by x - 1.

Related Topics

Polynomial Functions
Study the behavior and properties of polynomial functions and their graphs.
intermediate
Rational Functions
Explore functions that are ratios of polynomials and their characteristics.
intermediate
Algebraic Structures
Learn about algebraic systems and their properties, including rings and fields.
advanced
Complex Numbers
Understand the role of complex numbers in polynomial equations and their solutions.
intermediate

Key Concepts

Irreducible PolynomialsDegree of a PolynomialFactor TheoremPolynomial Division