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The FOIL method is a special case of a more general method for multiplying algebraic expressions using the distributive law. The word FOIL was originally intended solely as a mnemonic for high-school students learning algebra. The term appears in William Betz's 1929 text Algebra for Today, where he states: [2]
For binomial multiplication, distribution is sometimes referred to as the FOIL Method [2] (First terms , Outer , Inner , and Last ) such as: (+) (+) = + + +. In all semirings , including the complex numbers , the quaternions , polynomials , and matrices , multiplication distributes over addition: u ( v + w ) = u v + u w , ( u + v ) w = u w + v ...
[1] [2] When n = 2, it is easy to see why this is incorrect: (x + y) 2 can be correctly computed as x 2 + 2xy + y 2 using distributivity (commonly known by students in the United States as the FOIL method). For larger positive integer values of n, the correct result is given by the binomial theorem.
The article should simply say that FOIL is an acronym for a mnemonic with a basic description of what it refers to, then link to an exposition about multiplying polynomials which should lie elsewhere.
All the above multiplication algorithms can also be expanded to multiply polynomials. Alternatively the Kronecker substitution technique may be used to convert the problem of multiplying polynomials into a single binary multiplication. [30] Long multiplication methods can be generalised to allow the multiplication of algebraic formulae:
(To multiply more than 2 expressions, just multiply 2 at a time) To multiply two factors, each term of the first factor must be multiplied by each term of the other factor. If both factors are binomials, the FOIL rule can be used, which stands for "First Outer Inner Last," referring to the terms that are multiplied together. For example, expanding
Examples include a recent proposal to make over-the-counter birth control free, as well as Biden’s executive order in August 2022 to support people who travel out of state for an abortion. The ...
In mathematics, the polynomial method is an algebraic approach to combinatorics problems that involves capturing some combinatorial structure using polynomials and proceeding to argue about their algebraic properties. Recently, the polynomial method has led to the development of remarkably simple solutions to several long-standing open problems ...