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}

so

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}

Read more about this topic:  FOIL Method

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

    ““A sigh for every so many breath,
    And for every so many sigh a death.
    That’s what I always tell my wife
    Is the multiplication table of life.””
    —Robert Frost (1874–1963)

    “If you have abandoned one faith, do not abandon all faith. There is always an alternative to the faith we lose. Or is it the same faith under another mask?”
    —Graham Greene (1904–1991)

    “You have waited, you always wait, you dumb, beautiful ministers,
    We receive you with free sense at last, and are insatiate
    hence-forward,
    Not you any more shall be able to foil us, or withhold yourselves
    from us,
    We use you, and do not cast you aside—we plant you permanently within us,
    We fathom you not—we love you—there is perfection in you also,
    You furnish your parts, toward eternity,
    Great or small, you furnish your parts toward the soul.”
    —Walt Whitman (1819–1892)