Factoring special products is a key skill in algebra. It helps simplify complex expressions and solve equations. Perfect square trinomials, differences of squares, and sums/differences of cubes are common patterns you'll encounter.

Recognizing these patterns lets you quickly factor expressions that might seem tricky at first. Mastering these techniques will make solving equations and simplifying algebraic expressions much easier. They're essential tools in your algebra toolkit.

Factoring Special Products

Perfect square trinomial factoring

• Perfect square trinomials have the form $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$ where $a$ and $b$ are any algebraic expressions
• $a^2 + 2ab + b^2$ factors to $(a + b)^2$ because the square of a binomial follows the pattern of a perfect square trinomial with two positive terms ($a^2$ and $b^2$) and a middle term that is twice the product of $a$ and $b$
• $a^2 - 2ab + b^2$ factors to $(a - b)^2$ because the square of a binomial with a negative second term results in a perfect square trinomial with two positive terms ($a^2$ and $b^2$) and a negative middle term
• To factor perfect square trinomials:
  1. Identify the first term ($a^2$) and the last term ($b^2$) of the trinomial
  2. Find the square roots of the first term ($a$) and the last term ($b$)
  3. Check the sign of the middle term (positive or negative) to determine the sign in the binomial factor
     • If the middle term is positive, use $(a + b)^2$ ($x^2 + 6x + 9 = (x + 3)^2$)
     • If the middle term is negative, use $(a - b)^2$ ($x^2 - 10x + 25 = (x - 5)^2$)
• This process is related to completing the square, which is used to solve quadratic equations and convert them to vertex form

Differences of squares factoring

• Differences of squares have the form $a^2 - b^2$ where $a$ and $b$ are any algebraic expressions
• $a^2 - b^2$ factors to $(a + b)(a - b)$ because the product of the sum and difference of two terms results in a difference of squares ($x^2 - 9 = (x + 3)(x - 3)$)
• To factor differences of squares:
  1. Identify the two squared terms ($a^2$ and $b^2$)
  2. Find the square roots of each term ($a$ and $b$)
  3. Write the two binomial factors $(a + b)$ and $(a - b)$
     • The first factor is the sum of the square roots ($a + b$)
     • The second factor is the difference of the square roots ($a - b$) ($x^2 - 16 = (x + 4)(x - 4)$)
• The factors $(a + b)$ and $(a - b)$ are conjugates, which are useful in various algebraic operations

Sums and differences of cubes

• Sum of cubes has the form $a^3 + b^3$ where $a$ and $b$ are any algebraic expressions
  • $a^3 + b^3$ factors to $(a + b)(a^2 - ab + b^2)$ ($x^3 + 8 = (x + 2)(x^2 - 2x + 4)$)
• Difference of cubes has the form $a^3 - b^3$ where $a$ and $b$ are any algebraic expressions
  • $a^3 - b^3$ factors to $(a - b)(a^2 + ab + b^2)$ ($x^3 - 27 = (x - 3)(x^2 + 3x + 9)$)
• To factor sums and differences of cubes:
  1. Identify the two cubed terms ($a^3$ and $b^3$)
  2. Find the cube roots of each term ($a$ and $b$)
  3. Write the first factor based on the sign between the cubed terms
     • For sum of cubes, use $(a + b)$
     • For difference of cubes, use $(a - b)$
  4. Write the second factor using the appropriate pattern
     • For sum of cubes, use $a^2 - ab + b^2$
     • For difference of cubes, use $a^2 + ab + b^2$

Additional Factoring Techniques

• The FOIL method is used to multiply binomials and can be reversed to factor quadratic expressions
• When factoring doesn't work, the quadratic formula can be used to solve quadratic equations directly