Special Factorizations

Difference of Squares

$a^2 - b^2 = (a + b)(a - b)$

Examples

• $x^2 - 9 = (x + 3)(x - 3)$
• $4x^2 - 25 = (2x + 5)(2x - 5)$
• $x^4 - 16 = (x^2 + 4)(x^2 - 4) = (x^2 + 4)(x + 2)(x - 2)$

Perfect Square Trinomials

$a^2 + 2ab + b^2 = (a + b)^2$

$a^2 - 2ab + b^2 = (a - b)^2$

Examples

• $x^2 + 6x + 9 = (x + 3)^2$
• $x^2 - 10x + 25 = (x - 5)^2$
• $4x^2 + 12x + 9 = (2x + 3)^2$

Sum and Difference of Cubes

Sum of Cubes

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Difference of Cubes

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Memory Aid: SOAP

• Same sign as the original
• Opposite sign
• Always Positive (last term)

Examples

• $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$
• $x^3 - 27 = (x - 3)(x^2 + 3x + 9)$
• $8x^3 + 125 = (2x + 5)(4x^2 - 10x + 25)$

Factoring by Grouping

$ac + ad + bc + bd = a(c + d) + b(c + d) = (a + b)(c + d)$

Example

• $x^3 + 2x^2 + 3x + 6 = x^2(x + 2) + 3(x + 2) = (x^2 + 3)(x + 2)$

Trinomial Factoring

Form: $x^2 + bx + c$

Find factors of $c$ that add to $b$:

$x^2 + bx + c = (x + m)(x + n)$

where $mn = c$ and $m + n = b$

Form: $ax^2 + bx + c$ ($a \neq 1$)

Method: Find factors of $ac$ that add to $b$, then factor by grouping.

Example

• $2x^2 + 7x + 3$
• $ac = 6$, need factors that add to 7: 6 and 1
• $2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)$

Factoring Completely

Always check for:

1. Common factors (GCF)
2. Difference of squares
3. Perfect square trinomials
4. Sum/difference of cubes
5. Trinomial patterns
6. Grouping

Example

• $2x^4 - 32 = 2(x^4 - 16) = 2(x^2 + 4)(x^2 - 4) = 2(x^2 + 4)(x + 2)(x - 2)$