Math101 · Independent Ontario learning library
Math101Difference of Squares
A difference of squares is a binomial $a^2-b^2$ that factors as $(a-b)(a+b)$. The identity follows because the middle terms cancel during expansion.
This pattern makes factoring fast, explains conjugates, and supports equation solving, rational-expression simplification, and algebraic identities.
Intuition and core definition
A difference of squares is a binomial $a^2-b^2$ that factors as $(a-b)(a+b)$. The identity follows because the middle terms cancel during expansion. Both terms must be perfect squares and the operation between them must be subtraction; a sum of squares does not factor this way over the real numbers.
Notation, language, and conditions
$a$ and $b$ may themselves be monomials or expressions. For example, $9x^4-25=(3x^2)^2-5^2$. Factoring should continue: $x^4-16=(x^2-4)(x^2+4)=(x-2)(x+2)(x^2+4)$ over the reals.
Why this idea matters
A difference of squares factors through conjugate binomials because the opposite middle terms cancel during multiplication.
A dependable method
- Remove any greatest common factor first.
- Verify that exactly two terms remain with a subtraction sign.
- Write each term as a square, identifying $a$ and $b$.
- Form conjugate factors $(a-b)(a+b)$.
- Factor again if a factor is another difference of squares, then expand to check.
