Factoring polynomials is a crucial skill in algebra. It's all about breaking down complex expressions into simpler parts, making equations easier to solve. This process involves several methods, from finding common factors to recognizing special patterns.

Mastering these techniques opens doors to solving more advanced problems. Whether you're dealing with quadratic equations or simplifying complex expressions, factoring is a fundamental tool that will serve you well throughout your math journey.

General Strategy for Factoring Polynomials

Methods of polynomial factoring

• Greatest common factor (GCF): find the largest factor common to all terms and factor it out
• Difference of squares: $a^2 - b^2$ always factors as $(a+b)(a-b)$
  • Example: $x^2 - 9 = (x+3)(x-3)$
• Sum or difference of cubes: $a^3 \pm b^3$ always factors as $(a \pm b)(a^2 \mp ab + b^2)$
  • Example: $x^3 - 8 = (x-2)(x^2+2x+4)$
• Grouping: group terms, factor out common binomials, and multiply the results
• Trial and error (guess and check): guess factors based on the leading coefficient and constant term
• Perfect square trinomial: $a^2 + 2ab + b^2 = (a+b)^2$ or $a^2 - 2ab + b^2 = (a-b)^2$
  • Example: $x^2 + 6x + 9 = (x+3)^2$

Steps in general factoring strategy

1. Check for a greatest common factor (GCF) and factor it out

   • Find the GCF of the coefficients and variables (with the lowest exponent)
   • Factor out the GCF from each term in the polynomial
   • This process utilizes the distributive property

2. Identify the number of terms remaining in the polynomial

   • If two terms remain, attempt the appropriate two-term factoring method (difference of squares, sum/difference of cubes)
   • If three terms remain, attempt the appropriate three-term factoring method (grouping, trial and error, perfect square trinomial)
   • If four or more terms remain, attempt grouping or additional GCF factoring

3. Repeat steps 1 and 2 as necessary until the polynomial is fully factored

   • A polynomial is fully factored when no further factoring methods can be applied

Special cases of factoring

• Difference of squares: $a^2 - b^2$ always factors as $(a+b)(a-b)$
  • Examples:
    • $x^2 - 9 = (x+3)(x-3)$
    • $4x^2 - 25y^2 = (2x+5y)(2x-5y)$
• Perfect square trinomials: $a^2 + 2ab + b^2 = (a+b)^2$ or $a^2 - 2ab + b^2 = (a-b)^2$
  • The middle term is always twice the product of the square roots of the first and last terms
  • Examples:
    • $x^2 + 6x + 9 = (x+3)^2$
    • $x^2 - 10x + 25 = (x-5)^2$

Advanced Factoring Techniques

• Prime factorization: breaking down a polynomial into its prime factors
• Synthetic division: a shortcut method for dividing a polynomial by a linear factor
• Factorization by substitution: replacing a group of terms with a single variable to simplify the factoring process