Lecture Notes on Factoring and Solving Quadratics

Key Takeaways

• Main Objective: Understand that all trinomials can be factored into a product of two binomials.
• Efficiency in Factoring: The goal is to efficiently factor trinomials, especially given the simplicity of problems in this class.

Techniques for Factoring

• Distributive Property (FOIL Method):

  • Use the distributive property to understand how the product of two binomials creates a trinomial.
  • Example: For the trinomial with first term 2x², use binomials 2x and x.

• Finding Factors:

  • Identify numbers that multiply to give the last term in the trinomial.
  • Example: For a constant term of 2, options are 2 & 1 or -2 & -1. Positive middle term implies positive factors.

• Arranging Factors:

  • Ensure inner and outer terms combine to give the middle term.
  • Example setup where factors are adjusted to form 5x in the middle term by switching the order of numbers.

Solving Quadratics

• Beyond Factoring:

  • Factoring alone is not the solution; solving involves further steps.

• Zero Product Property:

  • If a product of factors equals zero, one or more factors must be zero.
  • Set each factor to zero and solve for the variable.
  • Example: From factored form (e.g., (x + 1)(x + 2) = 0), solve each: x + 1 = 0, x + 2 = 0.

• Final Steps:

  • Solve the equations derived from setting each factor equal to zero.
  • Remember to subtract and divide appropriately to find solutions to the quadratic.

Common Mistakes

• Incomplete Process:
  • Students often stop after factoring without applying the zero product property.
  • Ensure to solve for the variable to complete the problem.

Conclusion

• Review homework not just for correct factoring but for the complete solving process.
• Practice efficiency in factoring to prepare for more complex problems efficiently.