Concept Overview
Some polynomials follow special patterns that allow them to be factored quickly and efficiently. Recognizing these forms—such as squares or cubes—helps reveal useful structure. This page focuses on those recognizable patterns and how to apply the right identity for each.
Key Vocabulary
- Perfect square trinomial: A trinomial that is the square of a binomial.
- Difference of squares: An expression of the form \( a^2 – b^2 \), which factors into \( (a – b)(a + b) \).
- Sum or difference of cubes: Expressions like \( a^3 \pm b^3 \), which factor using specific binomial-trinomial patterns.
- Conjugate pair: Two binomials of the form \( (a – b)(a + b) \), which arise from factoring a difference of squares.
Common Factoring Identities
| Form | Factored Result | Conditions |
|---|---|---|
| \( a^2 – b^2 \) | \( (a – b)(a + b) \) | Both terms are perfect squares Subtraction only |
| \( a^2 \pm 2ab + b^2 \) | \( (a \pm b)^2 \) | Perfect square trinomial Middle term must be \( \pm 2ab \) |
| \( a^3 + b^3 \) | \( (a + b)(a^2 – ab + b^2) \) | Both terms are perfect cubes Use SOAP: Same, Opposite, Always Positive |
| \( a^3 – b^3 \) | \( (a – b)(a^2 + ab + b^2) \) | Both terms are perfect cubes Use SOAP |
Difference of Squares
A common pattern in algebra is the difference of squares, which follows the identity:
Key Identity
\( a^2 – b^2 = (a – b)(a + b) \)
This only applies when two perfect squares are subtracted. The expression must match the form “square minus square.”
The result is always the product of a conjugate pair: one binomial with addition and one with subtraction. This identity is foundational because it allows us to factor expressions that might not look factorable at first glance.
Example 1
Factor: \( x^2 – 16 \)
Recognize both terms are squares: \( x^2 = x^2 \), \( 16 = 4^2 \)
Apply the identity: \( x^2 – 4^2 = (x – 4)(x + 4) \)
Example 2
Factor: \( 9a^2 – 25b^2 \)
Both terms are perfect squares: \( (3a)^2 – (5b)^2 \)
Result: \( (3a – 5b)(3a + 5b) \)
Example 3 (Rewriting First)
Factor: \( 4x^2 – 49 \)
\( 4x^2 = (2x)^2 \), \( 49 = 7^2 \)
Result: \( (2x – 7)(2x + 7) \)
If the expression is a sum of squares, such as \( x^2 + 16 \), it cannot be factored using real numbers. There is no real-number identity for a sum of squares.
Perfect Square Trinomials
A perfect square trinomial is a quadratic expression that results from squaring a binomial. These trinomials follow predictable patterns and can be factored back into binomials using reverse recognition.
Key Identities
\( (a + b)^2 = a^2 + 2ab + b^2 \)
\( (a – b)^2 = a^2 – 2ab + b^2 \)
In each case, the first and last terms are perfect squares, and the middle term is twice the product of their square roots. Always check that the middle term equals \( 2ab \) (or \( -2ab \)), not just that the outer terms are squares.
Example 1
Factor: \( x^2 + 6x + 9 \)
\( x^2 = x^2 \), \( 9 = 3^2 \), and \( 6x = 2 \cdot x \cdot 3 \)
Recognized as \( (x + 3)^2 \)
Example 2
Factor: \( 4a^2 – 12a + 9 \)
\( 4a^2 = (2a)^2 \), \( 9 = 3^2 \), and \( -12a = 2 \cdot 2a \cdot -3 \)
Recognized as \( (2a – 3)^2 \)
Example 3 (Check Carefully)
Factor: \( x^2 + 10x + 25 \)
\( x^2 = x^2 \), \( 25 = 5^2 \), and \( 10x = 2 \cdot x \cdot 5 \)
Recognized as \( (x + 5)^2 \)
If the trinomial doesn’t exactly match the pattern—especially if the middle term isn’t twice the product of the square roots—it may still be factorable, but not as a perfect square. Use general trinomial methods instead.
Sum and Difference of Cubes
The sum or difference of two perfect cubes can be factored using special identities. These factorizations always produce a binomial times a trinomial.
Key Identities
\( a^3 + b^3 = (a + b)(a^2 – ab + b^2) \)
\( a^3 – b^3 = (a – b)(a^2 + ab + b^2) \)
These identities are often remembered using the acronym:
SOAP: Same, Opposite, Always Positive
• First sign: Same as original
• Second sign: Opposite
• Third sign: Always
positive
Example 1 (Sum of Cubes)
Factor: \( x^3 + 8 \)
\( x^3 = x^3 \), \( 8 = 2^3 \)
Apply identity: \( (x + 2)(x^2 – 2x + 4) \)
Example 2 (Difference of Cubes)
Factor: \( 27a^3 – 64 \)
\( 27a^3 = (3a)^3 \), \( 64 = 4^3 \)
Apply identity: \( (3a – 4)(9a^2 + 12a + 16) \)
Example 3 (Variable Expression)
Factor: \( 125x^3 + 343y^3 \)
\( 125x^3 = (5x)^3 \), \( 343y^3 = (7y)^3 \)
Apply identity: \( (5x + 7y)(25x^2 – 35xy + 49y^2) \)
These patterns only apply when both terms are perfect cubes. If not, other factoring strategies (such as factoring out a GCF first) may be necessary before applying these identities.
Try It Yourself!
- Factor completely: \( x^2 – 81 \)
- Factor: \( x^2 + 14x + 49 \)
- Factor: \( 8x^3 – 27 \)
Reveal Answers
- \( x^2 – 81 = (x – 9)(x + 9) \)
- \( x^2 + 14x + 49 = (x + 7)^2 \)
- \( 8x^3 – 27 = (2x – 3)(4x^2 + 6x + 9) \)
Connections & Extensions
These special forms connect directly to solving quadratic equations, graphing parabolas, and simplifying rational expressions. They’re also essential for calculus preparation—especially when integrating or simplifying complex expressions.