Factoring Polynomials: Methods and Concepts

Factoring Polynomials

Definition

• Factoring a polynomial means expressing it as a product of simpler expressions, called factors.
• Factoring is the reverse operation of multiplying polynomials.

Methods of Factoring

• Factoring out the greatest common factor (GCF)
  • Find the GCF of all terms in the polynomial.
  • Divide each term by the GCF.
• Factoring by grouping
  • Group terms with common factors.
  • Factor out the common factor from each group.
• Factoring quadratic expressions
  • Look for two binomials that multiply to the original quadratic expression.
  • Use the ac method or the reverse FOIL method to factor quadratic expressions.

Factoring Types

• Monomial factors
  • Factoring out the GCF of a polynomial.
• Binomial factors
  • Factoring a quadratic expression into two binomials.
• Trinomial factors
  • Factoring a cubic expression into three binomials.

Key Concepts

• Factoring by decomposition
  • Express a polynomial as a product of simpler polynomials.
• Irreducible polynomials
  • Polynomials that cannot be factored further.
• Prime polynomials
  • Irreducible polynomials that cannot be expressed as a product of simpler polynomials.

Tips and Tricks

• Look for common factors
  • Factor out the GCF of all terms in the polynomial.
• Use the distributive property
  • Expand and simplify the polynomial to reveal hidden factors.
• Check your work
  • Multiply the factors to ensure they equal the original polynomial.