Special Factoring Patterns
Special Factoring Patterns
Certain factoring patterns appear so frequently that recognizing them on sight saves enormous time. The difference of two squares, perfect square trinomials, and the sum and difference of cubes are the three major special patterns.
The difference of squares, a² − b² = (a + b)(a − b), is especially common and shows up everywhere from simplifying rational expressions to rationalizing denominators.
This lesson drills all three patterns so that you can identify and apply them instantly, freeing your attention for the harder parts of a problem.
Difference of Squares
$$a^2 - b^2 = (a + b)(a - b)$$
$x^2 - 49$
$(x + 7)(x - 7)$.
$4x^2 - 25$
$(2x)^2 - 5^2 = (2x + 5)(2x - 5)$.
$a^2 + b^2$ does not factor over the reals.
Perfect Square Trinomials
$$a^2 + 2ab + b^2 = (a + b)^2$$
$$a^2 - 2ab + b^2 = (a - b)^2$$
$x^2 + 10x + 25$
$x^2 + 2(5)x + 5^2 = (x + 5)^2$.
$9x^2 - 24x + 16$
$(3x)^2 - 2(3x)(4) + 4^2 = (3x - 4)^2$.
Sum and Difference of Cubes
$$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$
$$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$
Memory aid: SOAP — Same sign, Opposite sign, Always Positive.
$x^3 - 8$
$x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)$.
$27x^3 + 1$
$(3x)^3 + 1^3 = (3x + 1)(9x^2 - 3x + 1)$.
Factoring Strategy
- Factor out the GCF.
- Count terms: 2 → difference of squares/cubes; 3 → trinomial or perfect square; 4 → grouping.
- Check if each factor can be factored further.
Practice Problems
Show Answer Key
1. $(x+9)(x-9)$
2. $(4x+3)(4x-3)$
3. $(x+7)^2$
4. $(2x-5)^2$
5. $(x+3)(x^2 - 3x + 9)$
6. $(2x-1)(4x^2 + 2x + 1)$
7. $(x^2+4)(x+2)(x-2)$
8. $2(x+5)(x-5)$
9. $(x+4)(x^2 - 4x + 16)$
10. $(5x - 3)^2$
11. $3(x-2)(x^2+2x+4)$
12. $(x^2+1)(x+1)(x-1)$