FOIL Method - 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 products. The table equivalent to the FOIL rule looks like this.

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

In the case that these are polynomials, the terms of a given degree are found by adding along the antidiagonals

$\begin{matrix} \times & cx & d \\ ax & acx^2 & adx \\ b & bcx & bd \end{matrix}$

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

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

Similarly, to multiply one writes the same table

$\begin{matrix} \times & d & e & f & g \\ a & ad & ae & af & ag \\ b & bd & be & bf & bg \\ c & cd & ce & cf & cg \end{matrix}$

and sums along antidiagonals:

$\begin{align}(ax^2&+bx+c)(dx^3+ex^2+fx+g)\\ &= adx^5 + (ae+bd)x^4 + (af+be+cd)x^3 + (ag+bf+ce)x^2+(bg+cf)x+cg.\end{align}$

Famous quotes containing the words table, alternative and/or foil:

“The table kills more people than war does.”
—Catalan proverb, quoted in Colman Andrews, Catalan Cuisine.

“If the alternative is to keep all just men in prison, or give up war and slavery, the State will not hesitate which to choose.”
—Henry David Thoreau (1817–1862)

“Within his crib is surest ward,
This little babe will be thy guard,
If thou wilt foil thy foes with joy,
Then flit not from this heavenly boy.”
—Robert Southwell (1561?–1595)