Polynomial factorization

Polynomial factorization rewrites a polynomial as a product of simpler polynomials over a specified coefficient system. The available factors depend on whether coefficients are restricted to integers, rationals, reals, or complex numbers.

Factoring out the greatest common factor: A factoring method that removes the largest shared numerical or algebraic factor from every term. Removing shared factors is usually the first step before searching for deeper polynomial structure.
Polynomial: An expression formed from variables, coefficients, and nonnegative integer powers combined by addition and multiplication. Factorization applies to polynomials and preserves the expression as a whole.
Irreducible polynomial: A nonconstant polynomial that cannot be expressed as a product of lower-degree polynomials over a specified coefficient system. Irreducibility is the stopping point for factorization relative to the chosen coefficients.
Polynomial equation: An equation stating that a polynomial equals zero or another expression. Setting a factored polynomial equal to zero reduces solving to its individual factors.
Difference of squares: The identity a² − b² = (a − b)(a + b), which factors a difference of two squares. Recognizing two square terms turns certain quadratic and higher-degree polynomials into paired factors.
Polynomial ring: A ring whose elements are polynomials with coefficients from a specified ring. The coefficient ring sets which constants and polynomial factors are allowed.
Integer factorization: The decomposition of an integer into a product of integers, typically prime factors. Polynomial factors are expressions, not merely numerical divisors of coefficients.
Multiplicity of a root: The number of times a linear factor associated with a root occurs in a polynomial. Repeated factors reveal roots whose multiplicity affects graphs and algebraic calculations.
Factoring by grouping: A method that groups polynomial terms so each group yields a common factor, revealing a shared binomial. Grouping can expose a common factor when no single factor appears in every original term.
Polynomial root: A value at which a polynomial evaluates to zero. Each root yields a linear factor over a field containing that value.

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