Factoring special products helps you rewrite certain polynomials as products without using trial and error. These patterns matter because they appear often in algebra, graphing, solving equations, and simplifying expressions. When you recognize the structure of an expression, you can factor it quickly and accurately.
The main goal is to match the expression to a known form before doing any calculations.
Key Facts
- Difference of squares: a^2 - b^2 = (a - b)(a + b)
- Perfect-square trinomial: a^2 + 2ab + b^2 = (a + b)^2
- Perfect-square trinomial: a^2 - 2ab + b^2 = (a - b)^2
- Sum of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2)
- Difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2)
- Always factor out the greatest common factor first, such as 6x^2 - 24 = 6(x^2 - 4) = 6(x - 2)(x + 2)
Vocabulary
- Factor
- A factor is an expression that is multiplied by another expression to produce a given product.
- Difference of squares
- A difference of squares is a binomial in the form a^2 - b^2 that factors into (a - b)(a + b).
- Perfect-square trinomial
- A perfect-square trinomial is a trinomial that comes from squaring a binomial, such as (a + b)^2 or (a - b)^2.
- Sum of cubes
- A sum of cubes is a binomial in the form a^3 + b^3 that factors using the cube pattern.
- Greatest common factor
- The greatest common factor is the largest factor shared by all terms in an expression.
Common Mistakes to Avoid
- Factoring a^2 + b^2 as (a + b)(a - b) is wrong because the difference of squares pattern only works for subtraction, not addition.
- Forgetting the middle term in a perfect-square trinomial is wrong because (a + b)^2 equals a^2 + 2ab + b^2, not a^2 + b^2.
- Using the wrong signs for cubes is wrong because a^3 + b^3 factors as (a + b)(a^2 - ab + b^2), while a^3 - b^3 factors as (a - b)(a^2 + ab + b^2).
- Skipping the greatest common factor is wrong because the expression may not be fully factored, such as 3x^2 - 12 needing 3(x - 2)(x + 2).
Practice Questions
- 1 Factor completely: x^2 - 49.
- 2 Factor completely: 8x^3 + 27.
- 3 Explain how you can tell whether x^2 - 10x + 25 is a perfect-square trinomial before factoring it.