FOIL rule

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The FOIL rule, also sometimes known as the double distributive property or more colloquially as foiling, is commonly taught to U.S. high school students learning algebra, as a mnemonic for remembering how to multiply two binomials (polynomials with two terms). It is quite possibly one of the most popular mnemonic devices in the world of early high school–junior high mathematics,^{[citation needed]} along with SOH-CAH-TOA for remembering elementary trigonometric identities.

The name FOIL comes from the order in which one multiplies the terms of the binomials:

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 answer is then the sum of the terms obtained. Thus the general form is:

$(a+b)(c+d) = \underbrace{ac}_\mathrm{first} + \underbrace{ad}_\mathrm{outside} + \underbrace{bc}_\mathrm{inside} + \underbrace{bd}_\mathrm{last}$

Note that $a$ is both a "first" term and an "outer" term; $b$ is both a "last" and "inner" term, and so forth.

1 Examples
2 Proof
3 Tableau as an alternative to FOIL
4 Generalizations
5 References
6 See also

[edit] Examples

$(x+2y)(x+1) = x^2 + x + 2xy + 2y \,$
$(x+5)(x+7) = x^2 + 7x + 5x + 35 = x^2 + 12x + 35 \,$
$(x+1)(x-1) = x^2 - x + x - 1 = x^2 - 1 \,$

[edit] Proof

The FOIL rule can be shown to be equivalent to two applications of the distributive property.

$\begin{align} (a+b)(c+d) &{} = a(c+d) + b(c+d) \\ &{}= ac + ad + bc + bd. \end{align}$

[edit] Tableau 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. The table equivalent to the FOIL rule looks like this.

$\begin{matrix} \times & c & d \\ a & ac & ad \\ b & bc & bd \end{matrix}$

To multiply (a+b+c)(w+x+y+z), the table would be as follows.

$\begin{matrix} \times & w & x & y & z \\ a & aw & ax & ay & az \\ b & bw & bx & by & bz \\ c & cw & cx & cy & cz \end{matrix}$

The sum of the table entries is the product of the polynomials. Thus

$\begin{align} (a+b+c)(w+x+y+z) = {} & aw + ax + ay + az \\ & {} + bw + bx + by + bz \\ & {} + cw + cx + cy + cz . \end{align}$

[edit] 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 and recursion allows one to expand such products. For instance,

$\begin{align}(a+b+c+d)(x+y+z+w)=&\,((a+b)+(c+d))((x+y)+(z+w)) \\ =&\,(a+b)(x+y)+(a+b)(z+w) \\ &\,{}+(c+d)(x+y)+(c+d)(z+w) \\ =&\,ax+ay+bx+by+az+aw+bz+bw \\ &\,{}+cx+cy+dx+dy+cz+cw+dz+dw. \end{align}$

[edit] References

Steege, Ray & Bailey, Kerry (1997), Schaum's Outline of Theory and Problems of Intermediate Algebra, Schaum's Outline Series, New York: McGraw–Hill, p. 54, ISBN 978-0-07-060839-9