Factoring a Difference of Squares
Factoring a difference of squares is a technique where a perfect square subtracted from another perfect square is rewritten as two factors containing the same terms but opposite signs, following the formula a² - b² = (a + b)(a - b). This Grade 7 math skill is essential for simplifying algebraic expressions and solving equations. For example, 49x² - 16 can be recognized as (7x)² - 4², which factors into (7x + 4)(7x - 4). Students learn to identify perfect squares and apply this pattern, understanding that when multiplying the factors back using FOIL, the middle terms cancel out. Covered in OpenStax Algebra and Trigonometry Chapter 1: Prerequisites, this concept builds foundational algebra skills. Note that a sum of squares cannot be factored using this method.
Key Concepts
Property A difference of squares is a perfect square subtracted from a perfect square. It can be rewritten as two factors containing the same terms but opposite signs. $$a^2 b^2 = (a + b)(a b)$$ To factor, confirm both terms are perfect squares and write the factored form. A sum of squares cannot be factored.
Examples To factor $49x^2 16$, recognize this as $(7x)^2 4^2$. The factors are $(7x+4)(7x 4)$. To factor $100y^4 81z^2$, recognize this as $(10y^2)^2 (9z)^2$. The factors are $(10y^2+9z)(10y^2 9z)$. To factor $a^2 1$, recognize this as $a^2 1^2$. The factors are $(a+1)(a 1)$.
Explanation When you see a perfect square minus another perfect square, it factors into two identical binomials, one with a plus and one with a minus. This causes the middle terms from FOIL to cancel out, leaving just the first and last terms.
Common Questions
What is the difference of squares formula?
The difference of squares formula is a² - b² = (a + b)(a - b). This means any perfect square minus another perfect square can be factored into two binomials with the same terms but opposite signs.
How do you factor 49x² - 16 using difference of squares?
Recognize that 49x² - 16 equals (7x)² - 4². Using the formula, the factors are (7x + 4)(7x - 4).
Why do the middle terms cancel when factoring a difference of squares?
When you multiply (a + b)(a - b) using FOIL, the middle terms +ab and -ab cancel out, leaving only a² - b².
Can you factor a sum of squares like a² + b²?
No, a sum of squares cannot be factored using the difference of squares method. This formula only works when subtracting perfect squares.
How do you factor 100y⁴ - 81z² as a difference of squares?
Recognize this as (10y²)² - (9z)². The factors are (10y² + 9z)(10y² - 9z).
What is the factored form of a² - 1?
Since a² - 1 equals a² - 1², the factored form is (a + 1)(a - 1).