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:

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

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