Multiplying Polynomials The Easy Way

Multiplying Polynomials FOIL
To FOIL or Not to FOIL

Most students are familiar with the mnemonic FOIL to help them multiply two binomials (polynomials with 2 terms) together. As a simple example, we have

(x + 1)(x – 2) = x2 – 2x + x – 2 = x2 – x – 2

I do not particularly like teaching this method for multiplying binomials for the following reasons:

  1. This method works ONLY for binomials. It does not extend to polynomials with more than 2 terms.
  2. It requires learning something new. There is already an algorithm that works very well that simulates a method students already know.
  3. Most students after learning HOW to FOIL will not be able to explain WHY it works.
  4. From a theoretical point of view it is inefficient. FIOL is actually the better choice.

Some of you might be saying “But FOIL is so simple and easy to remember. What can possibly be better?” The answer is an algorithm that everyone already knows.

multiplying polynomials and numbers
Quick Review of Multiplication

As a quick review, let’s multiply two 2-digit numbers together, say 12 and 32:

12
32

We begin by multiplying the 2 on the bottom by each digit on top, moving from right to left:

12
32
24

We then multiply the 3 on the bottom by each digit on top, moving from right to left. This time as we write the answers we leave one blank space on the right:

12
32
24
36     

Finally, we add:

12
32
24
36     
384

Voila!

multiplying polynomials cute
A Better Algorithm For Multiplying Polynomials

We are now going to use essentially the same algorithm for multiplying our first two binomials above together. Notice that for multiplying numbers I chose an example where no carrying was necessary. When multiplying polynomials you will NEVER have to carry.

x + 1
x – 2

We begin by multiplying the -2 on the bottom by each term on top, moving from right to left. First note that -2 times 1 is -2:

x + 1
x – 2
     –2

Next note that -2 times x is -2x:

x + 1
x – 2
–2x – 2     

Now we multiply the x on the bottom by each term on top, moving from right to left. This time as we write the answers we leave one blank space on the right:

x + 1
x – 2
–2x – 2     
x2 + x               

Finally, we add:

x + 1
x – 2
–2x – 2     
x2 + x               
x2 – x – 2       

A “Harder” Example

One of the nicest things about this method of multiplying polynomials is that it can be used to multiply ANY two polynomials together  (not just binomials). For example, let’s multiply the trinomials x2 + 2x – 1 and 2x2 – x + 3 together.

    x2 + 2x – 1
    2x2  –   x + 3   
3x2 + 6x – 3  
–x3 – 2x2  +  x                     
2x4 + 4x3 – 2x2                                    
2x4 + 3x3  – x2  +   7x – 3                        

Summary and Further Exploration

To summarize, the algorithm I illustrated in this post is better than FOILING for the following reasons:

  1. This method works for multiplying ANY two polynomials together, whereas FOIL works ONLY for binomials.
  2. Everyone already essentially knows this algorithm from elementary school.
  3. It is much easier to see WHY this algorithm works. I’m going to leave the “why” as an exercise.

Let me leave you a few problems to practice on your own as well as some theoretical questions.

Exercises

  1. Multiply the polynomials 3x4 + 5x3 – x2 + x – 1 and 2x3 – 3x2 + x – 4.
  2. Explain why the algorithm presented here works. In particular, how is the distributive property being used?
  3. Show that FOIL is equivalent to the distributive property.
  4. Why is FIOL more efficient than FOIL? In other words which property of the real numbers do you need to FOIL that you DO NOT need to FIOL?

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See you tomorrow…

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