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The FOIL Method and Multiplying Polynomials

Date: 05/25/2008 at 11:32:46
From: Mohamed
Subject: multiplying using foil

Multiply using foil (d-4)(d+3).  How is it done?

Date: 05/25/2008 at 22:31:02
From: Doctor Peterson
Subject: Re: multiplying using foil

Hi, Mohamed.

The FOIL idea is one way to apply the distributive property to 
multiply two binomials.  Essentially what it means is that we need to
multiply EACH term in the first factor by EACH term in the second.  We
can list them like this:

  first times first:    d * d = d^2  [these are the First terms]
  first times second:   d * 3 = 3d   [these are the Outside terms]
  second times first:  -4 * d = -4d  [these are the Inside terms]
  second times second: -4 * 3 = -12  [these are the Last terms]

You can diagram the four multiplications this way:
     /___   \
    /    \   \
  (d - 4)(d + 3)
        \_/  /

The "outside" terms are the first and last; the "inside" terms are the
two in the middle.

Here is how I would actually write the work (including the step of
combining like terms):

  (d - 4)(d + 3) = d^2 + 3d - 4d - 12 = d^2 - d - 12

Notice a couple important things I did:

1. Although some people memorize the method as First, Outside, Inside,
Last, from which the method gets its name, you don't need to do so;
all you really need is "each times each" and an orderly way to get to
them all.  Then it works for ANY two polynomials, not just binomials.

2. When a term is subtracted rather than added, just include the
negative sign as part of the term, since "d - 4" is the same as "d +
-4".  That makes sure all the signs come out right.

3. In a typical problem like this one, you can expect the middle two
terms to combine; be careful with the signs!

When you have more than just binomials to multiply, a different way of
writing it, shown in the following page, can give an easier way to
organize your work:

  Explaining Algebra Concepts and FOIL 

A sort of hybrid method, which still writes everything in a line but
groups like terms naturally, looks like this for your example:

  (d - 4)(d + 3) = d^2 + 3d
                       - 4d - 12
                 = d^2 -  d - 12

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Polynomials

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