"In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain."
Factoring involves breaking down a polynomial into smaller, factorable components.
Factors: The numbers that are multiplied together to give a product are called factors.
Prime and Composite Numbers: A prime number is a number that is only divisible by 1 and itself, whereas Composite numbers are those that have more than two factors.
Factor Trees: A diagram that helps to find the prime factorization of a composite number.
Greatest Common Factor: GCF is the largest factor that divides two or more numbers without leaving a remainder.
Factoring out a Common Factor: Reducing an expression by factoring out the greatest common factor.
Difference of Squares: A polynomial that is written in the form a²-b² can be factored as (a+b)(a-b).
Perfect Square Trinomials: A trinomial that is the square of a binomial can be factored as (a+b)² and (a-b)².
Factorization by Grouping: Grouping terms in an expression and finding the common factor to simplify expressions.
Factoring by Substitution: A method of factoring quadratic expressions by substituting m and n so that the expression can be rewritten as the multiplication of two binomials.
Factoring Quadratic Trinomials: A method of factoring by splitting the middle term.
Sum of Cubes: A polynomial that is written in the form a³+b³ can be factored as (a+b)(a²-ab+b²).
Difference of Cubes: A polynomial that is written in the form a³-b³ can be factored as (a-b)(a²+ab+b²).
Rational Expressions and Factoring: Simplifying fractional expressions by factoring the numerator and the denominator.
Factoring Polynomials with four or more terms: Factoring expressions with more than three terms by grouping or other methods.
GCF Factoring: This involves finding the greatest common factor of terms in a polynomial and factoring it out.
Difference of Two Squares Factoring: This involves factoring a polynomial where two terms are perfect squares, and the sign between them is negative.
Trinomial Factoring: This involves factoring a polynomial into three terms, where the first and the last term are constants and the middle term is a variable.
Perfect Square Trinomial Factoring: This involves factoring a polynomial where the first and the last term are perfect squares, and the middle term is twice the product of their square roots.
Sum and Difference of Two Cubes Factoring: This involves factoring a polynomial into a sum or difference of two cubes.
Grouping Factoring: This involves grouping terms in a polynomial together to create a common factor.
Trial and Error Factoring: This involves guessing and checking different factor combinations until the polynomial is completely factored.
Quadratic Factoring: This involves factoring a polynomial of degree two into two linear factors.
Rational Root Theorem Factoring: This involves finding potential rational roots of a polynomial and testing them to factor the polynomial completely.
Synthetic Division Factoring: This involves using synthetic division to check for potential roots and to factor a polynomial.
"Polynomial factorization is one of the fundamental components of computer algebra systems."
"The first polynomial factorization algorithm was published by Theodor von Schubert in 1793."
"Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension."
"But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems."
"When the long-known finite step algorithms were first put on computers, they turned out to be highly inefficient."
"The fact that almost any uni- or multivariate polynomial of degree up to 100 and with coefficients of a moderate size (up to 100 bits) can be factored by modern algorithms in a few minutes of computer time indicates how successfully this problem has been attacked during the past fifteen years."
"Nowadays, modern algorithms and computers can quickly factor univariate polynomials of degree more than 1000 having coefficients with thousands of digits."
"For this purpose, even for factoring over the rational numbers and number fields, a fundamental step is a factorization of a polynomial over a finite field."
"The fact that almost any uni- or multivariate polynomial of degree up to 100 and with coefficients of a moderate size (up to 100 bits) can be factored by modern algorithms..."
"...polynomials of degree more than 1000 having coefficients with thousands of digits."
"expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain."
"The first polynomial factorization algorithm was published by Theodor von Schubert in 1793."
"Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension."
"The fact that almost any uni- or multivariate polynomial of degree up to 100 and with coefficients of a moderate size (up to 100 bits) can be factored by modern algorithms in a few minutes of computer time indicates how successfully this problem has been attacked during the past fifteen years."
"For this purpose, even for factoring over the rational numbers and number fields, a fundamental step is a factorization of a polynomial over a finite field."
"Nowadays, modern algorithms and computers can quickly factor univariate polynomials of degree more than 1000 having coefficients with thousands of digits."
"But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems."
"expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain."
"The fact that almost any uni- or multivariate polynomial of degree up to 100 and with coefficients of a moderate size (up to 100 bits) can be factored by modern algorithms in a few minutes of computer time indicates how successfully this problem has been attacked during the past fifteen years."