In mathematics, factorization (also factorisation in British English) or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x ¿ 4 factors as (x ¿ 2)(x + 2). In all cases, a product of simpler objects is obtained.
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Integer factorization
In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer. When the numbers are very large, no efficient, non-quantum integer factorization algorithm is known; an effort concluded in 2009 by several researchers factored a 232-digit number, utilizing hundreds of machines over a span of 2 years.
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3 (number)
3 is a number, numeral, and glyph. It is the natural number following 2 and preceding 4.
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Discriminant
In algebra, the discriminant of a polynomial is a function of its coefficients which gives information about the nature of its roots. For example, the discriminant of the quadratic polynomial is Here, if ¿ > 0, the polynomial has two real roots, if ¿ = 0, the polynomial has one real root, and if ¿ < 0, the polynomial has no real roots. The discriminant of the cubic polynomial is For higher degrees, the discriminant is always a polynomial function of the coefficients.
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Reduced ring
In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x = 0 implies x = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced. The nilpotent elements of a commutative ring A form an ideal of A, the so-called nilradical of A; therefore a commutative ring is reduced if and only if its nilradical is reduced to zero.
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Multiplication
Multiplication (often denoted by the cross symbol "×") is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic.
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Degree of a polynomial
The degree of a polynomial is the highest degree of its terms, when the polynomial is expressed in canonical form (i.e. as a linear combination of monomials). The degree of a term is the sum of the exponents of the variables that appear in it. The word degree is now standard, but in some older books, the word order may be used instead. For example, the polynomial has three terms. (Notice, this polynomial can also be expressed as .
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