FreeAlgebra                             Tutorials!  
Finding the Greatest Common Factor
Factoring Trinomials
Absolute Value Function
A Summary of Factoring Polynomials
Solving Equations with One Radical Term
Adding Fractions
Subtracting Fractions
The FOIL Method
Graphing Compound Inequalities
Solving Absolute Value Inequalities
Adding and Subtracting Polynomials
Using Slope
Solving Quadratic Equations
Multiplication Properties of Exponents
Completing the Square
Solving Systems of Equations by using the Substitution Method
Combining Like Radical Terms
Elimination Using Multiplication
Solving Equations
Pythagoras' Theorem 1
Finding the Least Common Multiples
Multiplying and Dividing in Scientific Notation
Adding and Subtracting Fractions
Solving Quadratic Equations
Adding and Subtracting Fractions
Multiplication by 111
Adding Fractions
Multiplying and Dividing Rational Numbers
Multiplication by 50
Solving Linear Inequalities in One Variable
Simplifying Cube Roots That Contain Integers
Graphing Compound Inequalities
Simple Trinomials as Products of Binomials
Writing Linear Equations in Slope-Intercept Form
Solving Linear Equations
Lines and Equations
The Intercepts of a Parabola
Absolute Value Function
Solving Equations
Solving Compound Linear Inequalities
Complex Numbers
Factoring the Difference of Two Squares
Multiplying and Dividing Rational Expressions
Adding and Subtracting Radicals
Multiplying and Dividing Signed Numbers
Solving Systems of Equations
Factoring Out the Opposite of the GCF
Multiplying Special Polynomials
Properties of Exponents
Scientific Notation
Multiplying Rational Expressions
Adding and Subtracting Rational Expressions With Unlike Denominators
Multiplication by 25
Decimals to Fractions
Solving Quadratic Equations by Completing the Square
Quotient Rule for Exponents
Simplifying Square Roots
Multiplying and Dividing Rational Expressions
Independent, Inconsistent, and Dependent Systems of Equations
Graphing Lines in the Coordinate Plane
Graphing Functions
Powers of Ten
Zero Power Property of Exponents
The Vertex of a Parabola
Rationalizing the Denominator
Test for Factorability for Quadratic Trinomials
Trinomial Squares
Solving Two-Step Equations
Solving Linear Equations Containing Fractions
Multiplying by 125
Exponent Properties
Multiplying Fractions
Adding and Subtracting Rational Expressions With the Same Denominator
Quadratic Expressions - Completing Squares
Adding and Subtracting Mixed Numbers with Different Denominators
Solving a Formula for a Given Variable
Factoring Trinomials
Multiplying and Dividing Fractions
Multiplying and Dividing Complex Numbers in Polar Form
Power Equations and their Graphs
Solving Linear Systems of Equations by Substitution
Solving Polynomial Equations by Factoring
Laws of Exponents
index casa mío
Systems of Linear Equations
Properties of Rational Exponents
Power of a Product and Power of a Quotient
Factoring Differences of Perfect Squares
Dividing Fractions
Factoring a Polynomial by Finding the GCF
Graphing Linear Equations
Steps in Factoring
Multiplication Property of Exponents
Solving Systems of Linear Equations in Three Variables
Solving Exponential Equations
Finding the GCF of a Set of Monomials
Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

Factoring Differences of Perfect Squares

Notice that

(u + v)(u – v) = (u + v)(u) + (u + v)(-v)

= (u)(u + v) + (-v)(u + v)

= (u)(u) + (u)(v) + (-v)(u) + (-v)(v)

= u 2 + uv – uv – v 2

= u 2 – v 2

So, an expression of the form u 2 – v 2 (called a difference of two perfect squares ) can always be factored into the form

u 2 – v 2 = (u + v)(u – v)

The formula in the box is a pattern. The symbols u and v may represent numbers, other algebraic symbols, or even algebraic expressions. It is the precise pattern which must be satisfied in each case.


Example 1:

Factor 4x 2 – y 2 .


First, we check, and find that the two terms contain no monomial factors in common. Then, we note that this is a two-term expression, so the methods for factoring trinomials that were demonstrated in detail in the previous document in the series are not likely to apply.

Whenever the expression to be factored contains just two terms, it is worth checking to see if the difference of squares pattern above can be used. You need to identify that the expression has two important features:

(i) it is a difference of two terms – the second term must be subtracted from the first term. In this example, we do have a difference of two terms.

(ii) it must be possible to write each of the two terms as a square of some expression. In this example, we see that

4x 2 = (2x) 2 and y 2 = (y) 2 ,

so this condition is met.

Matching the parts of this expression with the parts of the pattern in the box above, we get

4x 2 – y 2 = (2x) 2 – (y) 2 u 2 – v 2


u 2 = (2x) 2 giving u = 2x


v 2 = (y) 2 giving v = y


4x 2 – y 2 = u 2 – v 2 = (u + v)(u – v) = (2x + y)(2x – y)

or, directly

4x 2 – y 2 = (2x + y)(2x – y)

as the required factorization.

It is each to check this result directly.

(2x + y)(2x – y) = (2x + y)(2x) + (2x + y)(-y)

= (2x)(2x + y) + (-y)(2x + y)

= (2x)(2x) + (2x)(y) + (-y)(2x) + (-y)(-y)

= 4x 2 + 2xy – 2xy – y 2

= 4x 2 – y 2

Thus our factorization checks.


Example 2:

Factor 9a 4 – 36b 2 .


Again, we check that there are no common monomial factors in the two terms (there are none). Then, recognizing the possibility that this is a difference of two perfect squares (since this expression obviously is the difference of precisely two terms), we note that:

9a 4 = (3a 2 ) 2 and 36b 2 = (6b) 2

So, each of the two terms are perfect squares. So, we can apply the pattern formula given in the box above:

u 2 = (3a 2 ) 2 or u = 3a 2


v 2 = (6b) 2 or v = 6b

to get

9a 4 – 36b 2 = (3a 2 + 6b)(3a 2 – 6b)

You can easily verify by multiplication that this factorization is correct


Example 3:

Factor 9x 2 + 4y 2 .


This expression consists of two terms, both of which are perfect squares:

9x 2 + 4y 2 = (3x) 2 + (2y) 2

However, it is a sum of two perfect squares rather than a difference of two perfect squares. We might be tempted to try

9x 2 + 4y 2 ? (3x + 2y)(3x + 2y)

thinking that since a minus has become a plus on the left-hand side, perhaps the thing to do is to change the minus to a plus on the right-hand side. However, we should not simply assume this is the best we can do and move on, but rather, we must check to see if the assumption made here is correct. That is easy to do, as usual, by multiplication:

(3x + 2y)(3x + 2y) = (3x + 2y)(3x) + (3x + 2y)(2y)

= (3x)(3x + 2y) + (2y)(3x + 2y)

= (3x)(3x) + (3x)(2y) + (2y)(3x) + (2y)(2y)

= 9x 2 + 6xy + 6xy + 4y 2

= 9x 2 + 12xy + 4y 2

which is not the same as the original expression, 9x 2 + 4y 2 .

In fact, there is no way to factor the sum of two perfect squares. So, in answer to this example, we need to simply state that the given expression cannot be factored.

All Right Reserved. Copyright 2005-2024