7.4 Factor Special Products

Factoring special products is a key skill in algebra. It involves recognizing patterns in polynomials and using specific formulas to break them down into simpler expressions. This process is crucial for solving equations and simplifying complex algebraic expressions.

Mastering these techniques opens doors to more advanced math concepts. By learning to factor perfect square trinomials, sums and differences of cubes, and other special forms, you'll gain powerful tools for tackling a wide range of algebraic problems.

Factoring Special Products

Factoring perfect square trinomials

• Take the form $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$
  • $a^2$ and $b^2$ are perfect squares
  • $2ab$ is twice the product of the square roots of $a^2$ and $b^2$
• Can be factored as $(a + b)^2$ or $(a - b)^2$, respectively
  • Sign in the middle term determines the sign in the factored form
  • Square roots of $a^2$ and $b^2$ become the terms in the binomial factor
• Example: $x^2 + 6x + 9 = (x + 3)^2$
  • $x^2$ and $9$ are perfect squares, $6x$ is twice the product of their square roots
• These are examples of algebraic identities used in polynomial factorization

Sums and differences of cubes

• Sum of cubes takes the form $a^3 + b^3$
  • Can be factored as $(a + b)(a^2 - ab + b^2)$
  • First factor is the sum of the cube roots
  • Second factor is a quadratic expression with $a^2$, $-ab$, and $b^2$ terms
  • Example: $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$
• Difference of cubes takes the form $a^3 - b^3$
  • Can be factored as $(a - b)(a^2 + ab + b^2)$
  • First factor is the difference of the cube roots
  • Second factor is a quadratic expression with $a^2$, $ab$, and $b^2$ terms
  • Example: $x^3 - 27 = (x - 3)(x^2 + 3x + 9)$
• The factors $(a + b)$ and $(a - b)$ are conjugate expressions

Methods for complete polynomial factoring

1. Identify the type of polynomial

   • Quadratic ($ax^2 + bx + c$), cubic ($ax^3 + bx^2 + cx + d$), or higher degree
   • Look for special patterns like perfect square trinomials, differences of squares, sums or differences of cubes

2. Apply the appropriate factoring method

   • Common factors: Factor out the greatest common factor (GCF) first
   • Grouping: Group terms and factor out common binomials
   • Quadratic trinomials: Use trial and error or the ac method to find factors
   • Special products: Use formulas for perfect square trinomials, differences of squares, sums or differences of cubes

3. Repeat the process until the polynomial is completely factored

   • Continue factoring each factor if possible
   • Example: $2x^3 + 6x^2 - 8x = 2x(x^2 + 3x - 4) = 2x(x + 4)(x - 1)$

Recognition of special product patterns

• Perfect square trinomials
  • $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$
  • Middle term is twice the product of square roots of first and last terms
  • Example: $9x^2 + 12x + 4$ is a perfect square trinomial ($a=3x$, $b=2$)
• Differences of squares
  • $a^2 - b^2$
  • Two squared terms with a minus sign between them
  • Example: $25y^2 - 16$ is a difference of squares ($a=5y$, $b=4$)
• Sums of cubes
  • $a^3 + b^3$
  • Two cubed terms with a plus sign between them
  • Example: $8x^3 + 27$ is a sum of cubes ($a=2x$, $b=3$)
• Differences of cubes
  • $a^3 - b^3$
  • Two cubed terms with a minus sign between them
  • Example: $64m^3 - 125$ is a difference of cubes ($a=4m$, $b=5$)

Advanced Factoring Techniques

• Polynomial long division: Used to divide a polynomial by another polynomial of lower or equal degree
• Synthetic division: A shortcut method for dividing a polynomial by a linear factor $(x - r)$
• These techniques are useful when standard factoring methods are not applicable