FOIL method: Difference between revisions
Gatemansgc (talk | contribs) m Undid revision 1044385370 by 2600:1700:81A0:5F50:C0E1:365A:D47F:DA65 (talk) You don't need to put commentary in the article like that O_o;;; |
Reverting edit(s) by 2603:9000:E100:EC00:5CCA:C2EB:8F59:24CF (talk) to rev. 1169241203 by 65.92.110.142: Vandalism (RW 16.1) |
||
(19 intermediate revisions by 16 users not shown) | |||
Line 1: | Line 1: | ||
⚫ | |||
{{Short description|Mnemonic for finding the product of two binomial functions}} |
{{Short description|Mnemonic for finding the product of two binomial functions}} |
||
⚫ | |||
{{about|a mnemonic|other types of foil|Foil (disambiguation)}} |
|||
{{Infobox mathematical statement |
|||
⚫ | |||
| name = {{PAGENAMEBASE}} |
|||
| type = Method |
|||
| field = [[Elementary algebra]], [[elementary arithmetic]] |
|||
| statement = A technique for multiplying two [[Binomial (polynomial)|binomials]] in an [[algebraic expression]] using [[distributive law]]. |
|||
| image = MonkeyFaceFOILRule.JPG |
|||
⚫ | |||
| conjectured by = |
|||
| conjecture date = |
|||
| first stated by = William Betz |
|||
| first stated date = {{date and age|1929}} |
|||
}} |
|||
In [[elementary algebra]], '''''FOIL''''' is a [[mnemonic]] for the standard method of multiplying two [[Binomial (polynomial)|binomials]]<ref>{{cite web |url=https://www.wyzant.com/resources/lessons/math/algebra/foil | title=Simplifying using the FOIL Method Lessons | access-date=10 May 2018}}</ref>—hence the method may be referred to as the '''FOIL method'''. The word ''FOIL'' is an [[acronym]] for the four terms of the product: |
In [[elementary algebra]], '''''FOIL''''' is a [[mnemonic]] for the standard method of multiplying two [[Binomial (polynomial)|binomials]]<ref>{{cite web |url=https://www.wyzant.com/resources/lessons/math/algebra/foil | title=Simplifying using the FOIL Method Lessons | access-date=10 May 2018}}</ref>—hence the method may be referred to as the '''FOIL method'''. The word ''FOIL'' is an [[acronym]] for the four terms of the product: |
||
Line 19: | Line 29: | ||
... 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.) |
... 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.) |
||
</blockquote> |
</blockquote> |
||
William Betz was active in the |
William Betz was active in the movement to [[reform mathematics]] in the United States at that time, had written many texts on [[elementary mathematics]] topics and had "devoted his life to the improvement of mathematics education".<ref>{{citation |author=W. D. R. |title=Review of Algebra for Today : First Year |journal=The Mathematics Teacher |volume=30 |issue=7 |date=November 1937 |page=348 |publisher=National Council for the Teaching of Mathematics}}.</ref> |
||
Many students and educators in the |
Many students and educators in the US now use the word "FOIL" as a verb meaning "to expand the product of two binomials".<ref>{{Cite book |last=McCrea |first=Emma |url=https://books.google.com/books?id=IA-TDwAAQBAJ&dq=%22to+expand+the+product+of+two+binomials%22&pg=PT70 |title=Making Every Maths Lesson Count: Six principles to support great maths teaching (Making Every Lesson Count series) |date=2019-05-01 |publisher=Crown House Publishing Ltd |isbn=978-1-78583-421-9 |language=en}}</ref> |
||
==Examples== |
==Examples== |
||
Line 30: | Line 40: | ||
&= x^2 + 8x + 15. |
&= x^2 + 8x + 15. |
||
\end{align}</math> |
\end{align}</math> |
||
If either binomial involves [[subtraction]], the corresponding terms must be negated. For example, |
If either binomial involves [[subtraction]], the corresponding terms must be [[negated]]. For example, |
||
: <math>\begin{align} |
: <math>\begin{align} |
||
(2x - 3)(3x - 4) &= (2x)(3x) + (2x)(-4) + (-3)(3x) + (-3)(-4) \\ |
(2x - 3)(3x - 4) &= (2x)(3x) + (2x)(-4) + (-3)(3x) + (-3)(-4) \\ |
||
Line 38: | Line 48: | ||
==The distributive law== |
==The distributive law== |
||
{{See also|Distributive law}} |
|||
The FOIL method is equivalent to a two-step process involving the |
The FOIL method is equivalent to a two-step process involving the distributive law:<ref>{{Cite book |last1=Khare |first1=Apoorva |url=https://books.google.com/books?id=6jFOCgAAQBAJ&dq=FOIL+method++distributive+law:&pg=PA3 |title=Beautiful, Simple, Exact, Crazy: Mathematics in the Real World |last2=Lachowska |first2=Anna |date=2015 |publisher=Yale University Press |isbn=978-0-300-19089-2 |pages=3 |language=en |quote=This is sometimes called the “FOIL” method – essentially, it is simply the distributive law applied twice}}.</ref> |
||
: <math>\begin{align} |
: <math>\begin{align} |
||
(a + b)(c + d) &= a(c + d) + b(c + d) \\ |
(a + b)(c + d) &= a(c + d) + b(c + d) \\ |
||
&= ac + ad + bc + bd. |
&= ac + ad + bc + bd. |
||
\end{align}</math> |
\end{align}</math> |
||
In the first step, the ({{math|''c'' + ''d''}}) 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 |
In the first step, the ({{math|''c'' + ''d''}}) 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 distributivity can be applied easily to products with more terms such as [[trinomials]] and higher. |
||
==Reverse FOIL== |
==Reverse FOIL== |
||
The FOIL rule converts a |
The FOIL rule converts a product of two binomials into a sum of four (or fewer, if [[like terms]] are then combined) [[monomial]]s.<ref>{{Cite book |last1=Kirkland |first1=Carla C. |url=https://books.google.com/books?id=Om_NDwAAQBAJ&dq=Reverse+FOIL&pg=PA78 |title=Praxis Core For Dummies with Online Practice Tests |last2=Cleveland |first2=Chan |date=2020-01-29 |publisher=John Wiley & Sons |isbn=978-1-119-62047-1 |pages=78 |language=en |quote=...reverse FOIL can take you in the opposite direction from one expression to two-term expressions times each other. It's a form of factoring.}}</ref> The reverse process is called ''factoring'' or ''[[factorization]]''. In particular, if the proof above is read in reverse it illustrates the technique called [[Factorization#Factoring by grouping|factoring by grouping]]. |
||
==Table as an alternative to FOIL== |
==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 |
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 of multiplication. The table equivalent to the FOIL rule looks like this: |
||
: <math>\begin{array}{c|cc} |
: <math>\begin{array}{c|cc} |
||
\times & c & d \\ |
\times & c & d \\ |
||
Line 56: | Line 67: | ||
b & bc & bd |
b & bc & bd |
||
\end{array}</math> |
\end{array}</math> |
||
In the case that these are polynomials, {{math|(''ax'' + ''b'')(''cx'' + ''d'')}}, the terms of a given degree are found by adding along the [[antidiagonal]]s |
In the case that these are polynomials, {{math|(''ax'' + ''b'')(''cx'' + ''d'')}}, the terms of a given degree are found by adding along the [[antidiagonal]]s: |
||
: <math>\begin{array}{c|cc} |
: <math>\begin{array}{c|cc} |
||
\times & cx & d \\ |
\times & cx & d \\ |
||
Line 73: | Line 84: | ||
c & cw & cx & cy & cz |
c & cw & cx & cy & cz |
||
\end{array}</math> |
\end{array}</math> |
||
The sum of the table entries is the product of the polynomials. Thus |
The sum of the table entries is the product of the polynomials. Thus: |
||
:<math>\begin{align} |
:<math>\begin{align} |
||
(a + b + c)(w + x + y + z) &= (aw + ax + ay + az) \\ |
(a + b + c)(w + x + y + z) &= (aw + ax + ay + az) \\ |
||
Line 79: | Line 90: | ||
&+ (cw + cx + cy + cz). |
&+ (cw + cx + cy + cz). |
||
\end{align}</math> |
\end{align}</math> |
||
Similarly, to multiply {{math|(''ax''<sup>2</sup> + ''bx'' + ''c'')(''dx''<sup>3</sup> + ''ex''<sup>2</sup> + ''fx'' + ''g'')}}, one writes the same table |
Similarly, to multiply {{math|(''ax''<sup>2</sup> + ''bx'' + ''c'')(''dx''<sup>3</sup> + ''ex''<sup>2</sup> + ''fx'' + ''g'')}}, one writes the same table: |
||
:<math>\begin{array}{c|cccc} |
:<math>\begin{array}{c|cccc} |
||
\times & d & e & f & g \\ |
\times & d & e & f & g \\ |
||
Line 94: | Line 105: | ||
==Generalizations== |
==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 [[associativity|associative law]] and recursive foiling allows one to expand such products. For instance |
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 [[associativity|associative law]] and recursive foiling allows one to expand such products. For instance: |
||
: <math>\begin{align} |
: <math>\begin{align} |
||
(a + b + c + d)(x + y + z + w) &= ((a + b) + (c + d))((x + y) + (z + w)) \\ |
(a + b + c + d)(x + y + z + w) &= ((a + b) + (c + d))((x + y) + (z + w)) \\ |
||
Line 102: | Line 113: | ||
&+ cx + cy + dx + dy + cz + cw + dz + dw. |
&+ cx + cy + dx + dy + cz + cw + dz + dw. |
||
\end{align}</math> |
\end{align}</math> |
||
Alternate methods based on distributing forgo the use of the FOIL rule, but may be easier to remember and apply. For example |
Alternate methods based on distributing forgo the use of the FOIL rule, but may be easier to remember and apply. For example: |
||
:<math>\begin{align} |
:<math>\begin{align} |
||
(a + b + c + d)(x + y + z + w) &= (a + (b + c + d))(x + y + z + w) \\ |
(a + b + c + d)(x + y + z + w) &= (a + (b + c + d))(x + y + z + w) \\ |
Latest revision as of 01:49, 7 March 2024
This article needs additional citations for verification. (April 2016) |
Type | Method |
---|---|
Field | Elementary algebra, elementary arithmetic |
Statement | A technique for multiplying two binomials in an algebraic expression using distributive law. |
First stated by | William Betz |
First stated in | 1929 |
In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials[1]—hence the method may be referred to as the FOIL method. The word FOIL is an acronym 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
Note that a is both a "first" term and an "outer" term; b 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
[edit]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]
... 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 movement to reform mathematics in the United States at that time, had written many texts on elementary mathematics topics and had "devoted his life to the improvement of mathematics education".[3]
Many students and educators in the US now use the word "FOIL" as a verb meaning "to expand the product of two binomials".[4]
Examples
[edit]The method is most commonly used to multiply linear binomials. For example,
If either binomial involves subtraction, the corresponding terms must be negated. For example,
The distributive law
[edit]The FOIL method is equivalent to a two-step process involving the distributive law:[5]
In the first step, the (c + d) 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 distributivity can be applied easily to products with more terms such as trinomials and higher.
Reverse FOIL
[edit]The FOIL rule converts a product of two binomials into a sum of four (or fewer, if like terms are then combined) monomials.[6] The reverse process is called factoring or factorization. In particular, if the proof above is read in reverse it illustrates the technique called factoring by grouping.
Table as an alternative to FOIL
[edit]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 of multiplication. The table equivalent to the FOIL rule looks like this:
In the case that these are polynomials, (ax + b)(cx + d), the terms of a given degree are found by adding along the antidiagonals:
so
To multiply (a + b + c)(w + x + y + z), the table would be as follows:
The sum of the table entries is the product of the polynomials. Thus:
Similarly, to multiply (ax2 + bx + c)(dx3 + ex2 + fx + g), one writes the same table:
and sums along antidiagonals:
Generalizations
[edit]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 and recursive foiling allows one to expand such products. For instance:
Alternate methods based on distributing forgo the use of the FOIL rule, but may be easier to remember and apply. For example:
See also
[edit]References
[edit]- ^ "Simplifying using the FOIL Method Lessons". Retrieved 10 May 2018.
- ^ Betz, William (1929), Algebra for Today (vol. 1), Ginn and Company, p. 291.
- ^ W. D. R. (November 1937), "Review of Algebra for Today : First Year", The Mathematics Teacher, 30 (7), National Council for the Teaching of Mathematics: 348.
- ^ McCrea, Emma (2019-05-01). Making Every Maths Lesson Count: Six principles to support great maths teaching (Making Every Lesson Count series). Crown House Publishing Ltd. ISBN 978-1-78583-421-9.
- ^ Khare, Apoorva; Lachowska, Anna (2015). Beautiful, Simple, Exact, Crazy: Mathematics in the Real World. Yale University Press. p. 3. ISBN 978-0-300-19089-2.
This is sometimes called the "FOIL" method – essentially, it is simply the distributive law applied twice
. - ^ Kirkland, Carla C.; Cleveland, Chan (2020-01-29). Praxis Core For Dummies with Online Practice Tests. John Wiley & Sons. p. 78. ISBN 978-1-119-62047-1.
...reverse FOIL can take you in the opposite direction from one expression to two-term expressions times each other. It's a form of factoring.
Further reading
[edit]- Steege, Ray; Bailey, Kerry (1997). Schaum's Outline of Theory and Problems of Intermediate Algebra. Schaum's Outline Series. New York: McGraw–Hill. ISBN 978-0-07-060839-9.