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In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2) (x + 2) is a polynomial ...
The formula for the difference of two squares can be used for factoring polynomials that contain the square of a first quantity minus the square of a second quantity. For example, the polynomial x 4 − 1 {\displaystyle x^{4}-1} can be factored as follows:
The cube of a number or any other mathematical expression is denoted by a superscript 3, for example 23 = 8 or (x + 1)3 . The cube is also the number multiplied by its square : n3 = n × n2 = n × n × n. The cube function is the function x ↦ x3 (often denoted y = x3) that maps a number to its cube. It is an odd function, as.
Here the function is . In algebra, a cubic equation in one variable is an equation of the form. in which a is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation.
Solve for y using any method for solving such equations (e.g. conversion to a reduced cubic and application of Cardano's formula). Any of the three possible roots will do. Folding the second perfect square. With the value for y so selected, it is now known that the right side of equation is a perfect square of the form
Cabtaxi numbers are numbers that can be expressed as a sum of two positive or negative integers or 0 cubes in n ways. The smallest cabtaxi number, after Cabtaxi (1), is 91, [5] expressed as: or. The smallest Cabtaxi number expressed in 3 different ways is 4104, [6] expressed as. , or.
In algebra, a quartic function is a function of the form. α. where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial . A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form. where a ≠ 0. [1]
The quadratic sieve consists of computing the remainder of a2 / n for several a, then finding a subset of these whose product is a square. This will yield a congruence of squares. For example, consider attempting to factor the number 1649. We have: . None of the integers is a square, but the product is a square.