Factoring a Difference of Squares

It may be helpful to review these exercises first:

Recognizing Products and Sums; Identifying Factors and Terms talks about the concept of factoring
Writing Expressions in the Form $\,A^2$ helps you rename expressions as squares

Recall that factoring is the process of taking a sum/difference (things added/subtracted) and renaming it as a product (things multiplied).

An expression of the form $\,A^2 - B^2\,$ is called a difference of squares. It's a difference, because the last operation being performed is subtraction. It's a difference of squares, because both $\,A^2\,$ and $\,B^2\,$ are squares.

Using FOIL:

$$ \begin{align} &\cssId{s11}{(A + B)(A - B)}\cr &\qquad \cssId{s12}{= A^2 - AB + AB - B^2}\cr &\qquad \cssId{s13}{= A^2 - B^2} \end{align} $$

Thus, we have the following result:

FACTORING A DIFFERENCE OF SQUARES

For all real numbers $\,A\,$ and $\,B\,$: $$ \cssId{s17}{A^2 - B^2 = (A + B)(A - B)} $$

Examples

Factor: $x^2 - 4$

Solution:
$ \begin{align} &\cssId{s22}{x^2 - 4}\cr &\quad \cssId{s23}{= x^2 - 2^2}\cr &\quad \cssId{s24}{= (x + 2)(x - 2)} \end{align} $

Factor: $9 - x^2$

Solution:
$ \begin{align} &\cssId{s28}{9 - x^2}\cr &\quad \cssId{s29}{= 3^2 - x^2}\cr &\quad \cssId{s30}{= (3 + x)(3 - x)} \end{align} $

Factor: $4x^2 - 9$

Solution:
$ \begin{align} &\cssId{s34}{4x^2 - 9}\cr &\quad \cssId{s35}{= (2x)^2 - 3^2}\cr &\quad \cssId{s36}{= (2x + 3)(2x - 3)} \end{align} $

Factor: $49x^2 - 64y^2$

Solution:
$ \begin{align} &\cssId{s40}{49x^2 - 64y^2}\cr &\quad \cssId{s41}{= (7x)^2 - (8y)^2}\cr &\quad \cssId{s42}{= (7x + 8y)(7x - 8y)} \end{align} $

Factor: $x^6 - 25$

Solution:
$ \begin{align} &\cssId{s46}{x^6 - 25}\cr &\quad \cssId{s47}{= (x^3)^2 - 5^2}\cr &\quad \cssId{s48}{= (x^3 + 5)(x^3 - 5)} \end{align} $

Factor: $x^2 - 5$

Solution: Since $\,5\,$ is not a perfect square, this cannot be factored using integers. Note that it can be factored if we're allowed to use non-integers:
$ \begin{align} &\cssId{s54}{x^2 - 5}\cr &\quad \cssId{s55}{= x^2 - (\sqrt{5})^2}\cr &\quad \cssId{s56}{= (x + \sqrt{5})(x - \sqrt{5})} \end{align} $

In this exercise, you are factoring over the integers. That is, you are to use only the integers for your factoring. Recall that the integers are: $$\cssId{s60}{\{\ldots,\,-3,\,-2,\,-1,\,0,\,1,\,2,\,3,\,\ldots\}}$$

Factor: $x^2 + 4$

Solution: This can't be factored using integers. Usually, a sum of squares can't be factored.

(☆ The remaining discussion is beyond the scope of Algebra I; it is included for the benefit of more advanced readers.)

The expression $\,x^2 + 4\,$ can't even be factored using real numbers. It can be factored if we're allowed to use numbers that aren't real: $$ \begin{align} &\cssId{s69}{x^2 + 4}\cr &\quad \cssId{s70}{= x^2 - (2i)^2}\cr &\quad \cssId{s71}{= (x + 2i)(x - 2i)\,,} \end{align} $$ where $i^2 = -1\,.$ In general, a sum of squares can't be factored.

However, a sum of squares might also be a sum of cubes, which is factorable, like this:

$x^6 + 64$
$\ \ \cssId{s76}{= (x^3)^2 + 8^2}$	so, it's a sum of squares
$\ \ \cssId{s78}{= (x^2)^3 + 4^3}$	it's also a sum of cubes, which can be factored
$\ \ \cssId{s80}{= (x^2+4)(x^4 - 4x^2 + 16)}$

Use this: $$ A^3 + B^3 = (A + B)(A^2 - AB + B^2) $$

So, you can't just make a blanket statement that sums of squares aren't factorable.

Concept Practice

In this exercise, you are factoring over the integers. That is, you are to use only the integers for your factoring. Thus, for this exercise, not factorable means not factorable over the integers.