| 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 |
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:
The general form is
(a+b)(c+d)=\underbrace{ac}first+\underbrace{ad}outside+\underbrace{bc}inside+\underbrace{bd}last.
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]
The method is most commonly used to multiply linear binomials. For example,
\begin{align} (x+3)(x+5)&=x ⋅ x+x ⋅ 5+3 ⋅ x+3 ⋅ 5\\ &=x2+5x+3x+15\\ &=x2+8x+15. \end{align}
\begin{align} (2x-3)(3x-4)&=(2x)(3x)+(2x)(-4)+(-3)(3x)+(-3)(-4)\\ &=6x2-8x-9x+12\\ &=6x2-17x+12. \end{align}
See also: Distributive law. The FOIL method is equivalent to a two-step process involving the distributive law:[5]
\begin{align} (a+b)(c+d)&=a(c+d)+b(c+d)\\ &=ac+ad+bc+bd. \end{align}
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.
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:
\begin{array}{c|cc} x &c&d\\ \hline a&ac&ad\\ b&bc&bd \end{array}
\begin{array}{c|cc} x &cx&d\\ \hline ax&acx2&adx\\ b&bcx&bd \end{array}
(ax+b)(cx+d)=acx2+(ad+bc)x+bd.
To multiply, the table would be as follows:
\begin{array}{c|cccc} x &w&x&y&z\\ \hline a&aw&ax&ay&az\\ b&bw&bx&by&bz\\ c&cw&cx&cy&cz \end{array}
\begin{align} (a+b+c)(w+x+y+z)&=(aw+ax+ay+az)\\ &+(bw+bx+by+bz)\\ &+(cw+cx+cy+cz). \end{align}
\begin{array}{c|cccc} x &d&e&f&g\\ \hline a&ad&ae&af&ag\\ b&bd&be&bf&bg\\ c&cd&ce&cf&cg \end{array}
\begin{align} (ax2&+bx+c)(dx3+ex2+fx+g)\\ &=adx5+(ae+bd)x4+(af+be+cd)x3+(ag+bf+ce)x2+(bg+cf)x+cg. \end{align}
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:
\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}
\begin{align} (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)\\ & +(c+d)(x+y+z+w)\\ &=a(x+y+z+w)+b(x+y+z+w)\\ & +c(x+y+z+w)+d(x+y+z+w)\\ &=ax+ay+az+aw+bx+by+bz+bw\\ & +cx+cy+cz+cw+dx+dy+dz+dw. \end{align}