Factoring Methods

Factoring methods are essential tools in Algebra 1 that simplify polynomials and make solving equations easier. By mastering techniques like finding the GCF, grouping, and recognizing special forms, you can tackle a variety of polynomial expressions with confidence.

1. Greatest Common Factor (GCF)

   • Identify the largest factor that divides all terms in a polynomial.
   • Factor out the GCF to simplify expressions and make further factoring easier.
   • Use prime factorization to find the GCF of numerical coefficients.
   • The GCF can be a monomial, which can be factored out from each term.

2. Factoring by grouping

   • Group terms in pairs to find common factors within each group.
   • Factor out the common factor from each group to create a common binomial factor.
   • This method is particularly useful for polynomials with four or more terms.
   • Ensure that the resulting binomials are identical for successful factoring.

3. Factoring trinomials (a=1)

   • Look for a trinomial in the form of x² + bx + c.
   • Find two numbers that multiply to c and add to b.
   • Rewrite the trinomial as a product of two binomials: (x + p)(x + q).
   • This method is straightforward when the leading coefficient (a) is 1.

4. Factoring trinomials (a≠1)

   • Identify a trinomial in the form of ax² + bx + c where a ≠ 1.
   • Use the "ac method": multiply a and c, then find two numbers that multiply to ac and add to b.
   • Rewrite the middle term using the two numbers found, then factor by grouping.
   • The final result will be in the form of (px + q)(rx + s).

5. Factoring perfect square trinomials

   • Recognize the form a² ± 2ab + b², which factors to (a ± b)².
   • Check if the first and last terms are perfect squares and if the middle term is twice the product of the square roots.
   • This method simplifies expressions and is useful for solving equations.
   • Be mindful of the sign in the middle term when determining the correct binomial.

6. Factoring the difference of squares

   • Identify expressions in the form a² - b².
   • Use the formula a² - b² = (a + b)(a - b) to factor.
   • Both a and b must be perfect squares for this method to apply.
   • This technique is efficient for quickly simplifying and solving equations.

7. Factoring the sum and difference of cubes

   • Recognize the forms a³ + b³ and a³ - b³.
   • Use the formulas:
     • a³ + b³ = (a + b)(a² - ab + b²)
     • a³ - b³ = (a - b)(a² + ab + b²)
   • Ensure that both a and b are perfect cubes for accurate factoring.
   • This method is essential for solving higher-degree polynomials.