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In
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, ''FOIL'' is a
mnemonic A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding. Mnemonics make use of elaborative encoding, retrieval cues, and imag ...
for the standard method of multiplying two binomials—hence the method may be referred to as the FOIL method. The word ''FOIL'' is an
acronym An acronym is a word or name formed from the initial components of a longer name or phrase. Acronyms are usually formed from the initial letters of words, as in ''NATO'' (''North Atlantic Treaty Organization''), but sometimes use syllables, as ...
for the four terms of the product: * First ("first" terms of each binomial are multiplied together) * Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second) * Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second) * Last ("last" terms of each binomial are multiplied) The general form is : (a + b)(c + d) = \underbrace_\text + \underbrace_\text + \underbrace_\text + \underbrace_\text. Note that is both a "first" term and an "outer" term; is both a "last" and "inner" term, and so forth. The order of the four terms in the sum is not important and need not match the order of the letters in the word FOIL.


History

The FOIL method is a special case of a more general method for multiplying algebraic expressions using the
distributive law In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, ...
. The word ''FOIL'' was originally intended solely as a
mnemonic A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding. Mnemonics make use of elaborative encoding, retrieval cues, and imag ...
for high-school students learning algebra. The term appears in William Betz's 1929 text ''Algebra for Today'', where he states:
... first terms, outer terms, inner terms, last terms. (The rule stated above may also be remembered by the word FOIL, suggested by the first letters of the words first, outer, inner, last.)
William Betz was active in the mathematics reform movement in the
United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
at that time, had written many texts on elementary mathematics topics and had "devoted his life to the improvement of mathematics education". Many students and educators in the United States now use the word "FOIL" as a
verb A verb () is a word (part of speech) that in syntax generally conveys an action (''bring'', ''read'', ''walk'', ''run'', ''learn''), an occurrence (''happen'', ''become''), or a state of being (''be'', ''exist'', ''stand''). In the usual descri ...
meaning "to expand the product of two binomials".


Examples

The method is most commonly used to multiply
linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
binomials. For example, : \begin (x + 3)(x + 5) &= x \cdot x + x \cdot 5 + 3 \cdot x + 3 \cdot 5 \\ &= x^2 + 5x + 3x + 15 \\ &= x^2 + 8x + 15. \end If either binomial involves
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
, the corresponding terms must be negated. For example, : \begin (2x - 3)(3x - 4) &= (2x)(3x) + (2x)(-4) + (-3)(3x) + (-3)(-4) \\ &= 6x^2 - 8x - 9x + 12 \\ &= 6x^2 - 17x + 12. \end


The distributive law

The FOIL method is equivalent to a two-step process involving the
distributive law In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, ...
: : \begin (a + b)(c + d) &= a(c + d) + b(c + d) \\ &= ac + ad + bc + bd. \end In the first step, the () is distributed over the addition in first binomial. In the second step, the distributive law is used to simplify each of the two terms. Note that this process involves a total of three applications of the distributive property. In contrast to the FOIL method, the method using distributive can be applied easily to products with more terms such as trinomials and higher.


Reverse FOIL

The FOIL rule converts a
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of two binomials into a sum of four (or fewer, if
like terms In mathematics, like terms are summands in a sum that differ only by a numerical factor. Like terms can be regrouped by adding their coefficients. Typically, in a polynomial expression, like terms are those that contain the same variables to t ...
are then combined)
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer exponent ...
s. The reverse process is called ''factoring'' or ''
factorization In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a p ...
''. In particular, if the proof above is read in reverse it illustrates the technique called factoring by grouping.


Table as an alternative to FOIL

A visual memory tool can replace the FOIL mnemonic for a pair of polynomials with any number of terms. Make a table with the terms of the first polynomial on the left edge and the terms of the second on the top edge, then fill in the table with
products Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
. The table equivalent to the FOIL rule looks like this: : \begin \times & c & d \\ \hline a & ac & ad \\ b & bc & bd \end In the case that these are polynomials, , the terms of a given degree are found by adding along the
antidiagonal In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matrix. ...
s : \begin \times & cx & d \\ \hline ax & acx^2 & adx \\ b & bcx & bd \end so (ax + b)(cx + d) = acx^2 + (ad + bc)x + bd. To multiply , the table would be as follows: : \begin \times & w & x & y & z \\ \hline a & aw & ax & ay & az \\ b & bw & bx & by & bz \\ c & cw & cx & cy & cz \end The sum of the table entries is the product of the polynomials. Thus :\begin (a + b + c)(w + x + y + z) &= (aw + ax + ay + az) \\ &+ (bw + bx + by + bz) \\ &+ (cw + cx + cy + cz). \end Similarly, to multiply , one writes the same table :\begin \times & d & e & f & g \\ \hline a & ad & ae & af & ag \\ b & bd & be & bf & bg \\ c & cd & ce & cf & cg \end and sums along antidiagonals: : \begin (ax^2 &+ bx + c)(dx^3 + ex^2 + fx + g) \\ &= adx^5 + (ae + bd)x^4 + (af + be + cd)x^3 + (ag + bf + ce)x^2 + (bg + cf)x + cg. \end


Generalizations

The FOIL rule cannot be directly applied to expanding products with more than two multiplicands or multiplicands with more than two summands. However, applying the
associative law In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
and recursive foiling allows one to expand such products. For instance, : \begin (a + b + c + d)(x + y + z + w) &= ((a + b) + (c + d))((x + y) + (z + w)) \\ &= (a + b)(x + y) + (a + b)(z + w) \\ &+ (c + d)(x + y) + (c + d)(z + w) \\ &= ax + ay + bx + by + az + aw + bz + bw \\ &+ cx + cy + dx + dy + cz + cw + dz + dw. \end Alternate methods based on distributing forgo the use of the FOIL rule, but may be easier to remember and apply. For example, :\begin (a + b + c + d)(x + y + z + w) &= (a + (b + c + d))(x + y + z + w) \\ &= a(x + y + z + w) + (b + c + d)(x + y + z + w) \\ &= a(x + y + z + w) + (b + (c + d))(x + y + z + w) \\ &= a(x + y + z + w) + b(x + y + z + w) \\ &\qquad + (c + d)(x + y + z + w) \\ &= a(x + y + z + w) + b(x + y + z + w) \\ &\qquad + c(x + y + z + w) + d(x + y + z + w) \\ &= ax + ay + az + aw + bx + by + bz + bw \\ &\qquad + cx + cy + cz + cw + dx + dy + dz + dw. \end


See also

*
Binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
*
Factorization In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a p ...


References


Further reading

* {{DEFAULTSORT:Foil Method Elementary algebra Mnemonic acronyms Science mnemonics Multiplication