Factoring
Difference of Two Perfect Squares
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This expression is called a difference of two squares.
(Notice the subtraction sign between the terms.)

 You may remember seeing expressions like this one when you worked with multiplying algebraic expressions.  Do you remember ...
 

If you remember this fact, then you already know that:

 The factors of 

are

and

 

Remember:
An algebraic term is a perfect square when the numerical coefficient (the number in front of the variables) is a perfect square and the exponents of each of the variables are even numbers.

 

Example 1:

Factor:   x2 - 9

Both x2 and 9 are perfect squares.  Since subtraction is occurring between these squares, this expression is the difference of two squares.

What times itself will give x2 ?  The answer is x.
What times itself will give 9 ?  The answer is 3.

These answers could also be negative values, but positive values will make our work easier.

The factors are (x + 3) and (x - 3).
Answer:  (x + 3) (x - 3)  or   (x - 3) (x + 3)  
(order is not important)

 

Example 2:

Factor 4y2 - 36y6 

There is a common factor of 4y2 that can be factored out first in this problem, to make the problem easier.
                                         4y2 (1 - 9y4)
In the factor (1 - 9y4), 1 and 9y4 are perfect squares (their coefficients are perfect squares and their exponents are even numbers). 
Since subtraction is occurring between these squares, this expression is the difference of two squares.

What times itself will give 1?  The answer is 1.
What times itself will give 9y4 ?  The answer is 3y2 .
The factors are (1 + 3y2) and (1 - 3y2).
Answer:  4y2 (1 + 3y2) (1 - 3y2 or  4y2 (1 - 3y2) (1 + 3y2)

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If you did not see the common factor, you can begin with observing the perfect squares.  Both 4y2 and 36y6 are perfect squares (their coefficients are perfect squares and their exponents are even numbers).  Since subtraction is occurring between these squares, this expression is the difference of two squares.

What times itself will give 4y2 ?  The answer is 2y.
What times itself will give 36y6 ?  The answer is 6y3 .

The factors are (2y + 6y3) and (2y - 6y3).
Answer:  (2y + 6y3) (2y - 6y3 or  (2y - 6y3) (2y + 6y3
These answers can be further factored as each contains a common factor of 2y:
      2y (1 + 3y2) 2y (1 - 3y2) = 4y2 (1 + 3y2) (1 - 3y2

 

 

Be careful!!
This process of factoring does NOT apply to 
a 2 + b 2.

 

See how to use your
TI-83+/84+ graphing calculator  with factoring.
Click calculator.