Multiplying Polynomials

Date: 01/29/2001 at 17:30:52
From: Max Teodorescu
Subject: Multiplying a polynomial by a polynomial

Why isn't (3x+2) squared equal to 9x squared + 4?

If the 3x were squared, wouldn't it be equal to 9x squared, because 
3*3 = 9 and x*x = x squared?

Thanks for your time.

Date: 01/29/2001 at 19:05:22
From: Doctor Greenie
Subject: Re: Multiplying a polynomial by a polynomial

Hi, Max --

I always found it helpful to remember rules of multiplication by 
thinking of finding the area of rectangles. When I was very young, I 
learned my multiplication tables by setting up rows of marbles, or 
dominoes, or whatever, and counting the total number. For example:

    X X X X X X
    X X X X X X
    X X X X X X
    X X X X X X

Hmm - 4 rows with 6 in each row, and altogether there are 24, so 
4*6 = 24 (and also 6*4 = 24).

I can do the same thing by thinking of a rectangle without actually 
using the dominoes:

         9
    +-----------+
    |           |
    |           | 7
    |           |
    |           |
    +-----------+

This rectangle is 9 units by 7 units, so its area is 9*7 = 63 square 
units.

Now, what if I have something like 12*12? Perhaps you already know 
that 12*12 = 144; let's see a picture of it using a rectangle.  
(Actually, since the numbers we are multiplying together are the same, 
this will be a square.) Instead of writing the multiplication problem 
as "12*12", I will write it as "(10+2)*(10+2)," and I will draw my 
picture that way:

          10        2
    +-------------+---+
    |             |   |
    |             |   |
    |      A      | C |
    |             |   | 10
    |             |   |
    |             |   |
    +-------------+---+
    |      D      | B | 2
    +-------------+---+

Now if it were true that (10+2) squared were equal to (10) squared 
plus (2) squared, then the answer would be 100+4 = 104. Well, the 
(10) squared = 100 is the large square "A" in the picture, and the 
(2) squared = 4 is the small square "B" in the picture. But the 
picture for (10+2) squared includes not only the two squares A and B, 
but also the two rectangular regions C and D. Each of these rectangles 
has area 10*2 = 20, so the correct answer to 12*12 is (as you know) 
100+4+20+20 = 144.

It's the same thing with (3x+2) squared. To draw a picture of (3x+2) 
squared, we can just repeat the figure above with each 10 replaced by 
3x:

          3x        2
    +-------------+---+
    |             |   |
    |             |   |
    |      A      | C |
    |             |   | 3x
    |             |   |
    |             |   |
    +-------------+---+
    |      D      | B | 2
    +-------------+---+
  
Here, the area of region A is (3x) squared = 9x^2 (^2 means squared) 
and the area of region b is (2) squared = 4.  So your answer 9x^2+4 
represents the area of the two squares A and B. But (3x+2) squared 
also includes rectangles C and D, and the area of each of these 
rectangles is (3x)*(2) = 6x, so the area of the large square (and so 
the answer to (3x+2) squared) is
 
    9x^2 + 4 + 6x + 6x
or
    9x^2 + 12x + 4

I hope this picture helps.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/