CLOates

Revised:  23-Feb-2006

Multiplying Binomial Algebraic Expressions
A Geometric Interpretation

 

Using the following symbols:

 


o    unit square

 

 

 


       o    x-by-1 rectangle                          - or -

 

 

 

 


       o    x-by-x square                       

 

 

 

the binomial product (x + 3) (x + 2 ) can be represented as

 

 

 

 

 

 

 

 

 

 

 

 

 

 


By counting the above areas we find that

 

                                  (x + 3) (x + 2)  =  x2 + 5x + 6 .

 

 

QUESTIONS                 

1)  What binomial product is represented by diagram below?

 

 

 

 

 

 

 

 

 

 

 

 


                   Answer:  (x + 2) (2x + 4)

 

2)  Perform the indicated multiplication above and verify that the product corresponds to the above diagram.

 

 

 

 

 

 

 

         

Answer:   ___ x2    +   ____ x    +   ____

 

3)  Draw a loop around the squares that represent the first term, 2x2.  Label these squares TERM-1.

 

4)  Draw a loop around the rectangles that represent 8x.  Label these rectangles TERM-2.

 

5)  Circle the unit squares the represent 8 units.  Label these  squares TERM-3.


CLOates

Revised:  9-Oct-2005

Powers of Binomial Expressions

A Geometric Interpretation

 

It is possible to make geometric interpretations of such expressions as
(a + b), (a + b)2, and (a + b)
3 .  These may help you to visualize and, thereby, discover the meaning of such expressions.

 

1)  First Power, (a + b) :  length

 

Text Box: a                                                            a + b

 

 

2)  Second power, (a + b)2 :  area

 

 


                                                            (a + b) 2  =  a2 + 2ab + b2

 

Text Box: b
Text Box: b
Text Box: a
Text Box: b
 

 

 

 

 


3)  Third power, (a + b)3 :  volume

                                                            (a + b)3   =   a3  +  3a2b  +  3ab2  +  b3

Text Box: aText Box: bText Box: aText Box: aText Box: bText Box: bText Box: aText Box: aText Box: bText Box: bText Box: bText Box: a                                                         

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Disassembled Velveeta Model of Binomial Cube

 

 

The picture above shows the binomial cube (a + b)3 modeled with a cube of a popular cheese product.  The cube has been cut along the narrow lines pictured in 3) above, and disassembled into the constituent components a3  +  3a2b  +  3ab2  +  b3.

 

 

Pascal’s Triangle

 

While we’re on the topic of powers of binomial expressions, it would be useful to notice that the coefficients of the expanded expressions above for (a + b)0 [= 1], (a + b)1, (a + b)2, (a + b)3, (a + b)4, etc. form the pattern below.

 

1

1  1

1  2  1

1  3  3  1

1  4  6  4  1

1  5  10 10 5  1

 

Starting at the third row of the triangular pattern, notice that each number is the sum of the two numbers directly above it.  This pattern, first noticed by the French mathematician Blaise Pascal (1623 – 1662), is known as Pascal’s Triangle.  From the rows of the Triangle, we can determine without computation that, for instance,

 

(a + b)3   =   a3  +  3a2b  +  3ab2  + b3

 

and

 

(a + b)5   =   a5  +  5a4b  +  10 a3b2  +  10 a2b3  +  5 ab4  +  b5

 

That information could be handy for a bonus question on the next test.  Hint, hint.