Math A

Factoring
Difference of Two Perfect Squares

 

This expression is called a difference of two squares.

 

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 first in this problem.
                                         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 do 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 
a2 + b2.

 

How to use your
TI-83+ graphing calculator  with factoring.
Click calculator.


 
 

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Roberts