Factoring Polynomials Lecture Notes

Introduction to Factoring

• Purpose: Simplify polynomials to make solving equations easier.
• First Step: Always start with identifying the Greatest Common Factor (GCF).
• Benefits: Reduces the size of numbers, making further factoring easier.

Strategies for Different Types of Polynomials

Two-Term Polynomials

1. Difference of Two Squares:
   • Form: (a^2 - b^2 = (a + b)(a - b))
   • Example: (x^2 - 100) factors to ((x + 10)(x - 10))
2. Difference or Sum of Two Cubes:
   • Formulas:
     • Difference: (a^3 - b^3 = (a - b)(a^2 + ab + b^2))
     • Sum: (a^3 + b^3 = (a + b)(a^2 - ab + b^2))
   • Use "SOAP" (Same, Opposite, Always Positive) for signs.
   • Examples:
     • (x^3 - 8 = (x - 2)(x^2 + 2x + 4))
     • (2x^3 + 54 = 2(x + 3)(x^2 - 3x + 9))

Three-Term Polynomials (Trinomials)

1. Leading Coefficient = 1
   • Multiply last term to add to middle coefficient.
   • Example: (x^2 + 7x + 12) factors to ((x + 3)(x + 4))
2. Leading Coefficient (\neq 1)
   • Multiply the leading coefficient and the constant term.
   • Split the middle term and use factoring by grouping.
   • Example: (6x^2 - 11x - 10)
     • Multiply: (6 \times -10 = -60)
     • Split: (-15x + 4x)
     • Factor: ((3x + 2)(2x - 5))

Four-Term Polynomials

• Factoring by Grouping
  • Separate terms into two groups.
  • Factor the GCF from each group.
  • Example: (x^3 - 2x^2 - 4x + 8)
    • Group: ((x^3 - 2x^2) + (-4x + 8))
    • Factor: (x^2(x - 2) - 4(x - 2))
    • Final: ((x - 2)(x^2 - 4) = (x - 2)^2(x + 2))

Key Points to Remember

• Check for GCF first in any polynomial.
• Factoring is similar to reducing fractions; continue until the expression can no longer be factored.
• Some polynomials might be prime and cannot be factored further.
• Factoring by grouping is useful for four-term polynomials.

Conclusion

• Mastering these basic types of factoring will prepare you for more advanced factoring techniques.