Definition

1. Verb:
   • To resolve (a polynomial) into factors: To express a polynomial as a product of its factors, which are polynomials of lower degree. This is a fundamental operation in algebra.

Usage

• The verb factorize is used primarily in mathematics, specifically in algebra. It describes the process of breaking down a complex polynomial expression into simpler multiplicative components (factors).
• It is a transitive verb, requiring a direct object (the polynomial to be factorized).
• Common contexts include solving equations, simplifying expressions, and finding roots.

Examples

• Verb:
  • The student learned how to factorize quadratic equations.
  • Can you factorize the expression x² - 5x + 6?
  • The first step is to factorize the polynomial completely.

Advanced Usage

• "To factorize over the integers/rationals/reals": Specifies the number system (the set of numbers) within which the factors must be expressed.
  • The polynomial x² - 2 cannot be factorized over the rational numbers, but it can be factorized over the real numbers.
• "To factorize completely": To break down the polynomial into irreducible factors (factors that cannot be broken down further within a given number system).
  • Your final answer must show the polynomial factorized completely.

Variants and Related Words

• Factorization (n): The process or result of factorizing.
  • The factorization of x² - y² is (x - y)(x + y).
• Factor (n/v): As a noun, a component that multiplies with others; as a verb, a more general term meaning to resolve into factors, often used interchangeably with .
  • One factor of 12 is 3. (n)
  • We need to factor this expression. (v)

Synonyms

• Factor (v): To resolve into factors. (Note: In many contexts, and are synonyms, though is often preferred for polynomials.)
• Resolve into factors: A more descriptive phrase with the same meaning.

Related Phrases (Phrasal Verbs)

(Not commonly applicable for this technical mathematical term.)

Related Idioms

(Not commonly applicable for this technical mathematical term.)

Verb

1. resolve (a polynomial) into factors