Factoring a polynomial is a method used to break down the polynomial into simpler terms (factors) that, when multiplied together, give back the original polynomial.

Here’s a general approach to factoring different types of polynomials:

1. Factor Out the Greatest Common Factor (GCF)

• Step 1: Identify the greatest common factor among the coefficients and variables in all terms of the polynomial.
• Step 2: Factor out the GCF from each term.

2. Factoring by Grouping (for polynomials with four or more terms)

• Step 1: Group terms that have common factors.
• Step 2: Factor out the common factor from each group.
• Step 3: If the remaining terms inside the parentheses are the same, factor them out.

3. Factoring Trinomials

• For trinomials of the form [latex]ax^2+bx+c[/latex]:
  • Step 1: Look for two numbers that multiply to [latex]ac[/latex] (the product of the coefficient of [latex]x^2[/latex] and the constant term) and add to [latex]b[/latex] (the coefficient of [latex]x[/latex]).
  • Step 2: Use these numbers to split the middle term and factor by grouping.

4. Factoring Differences of Squares

• For expressions like [latex]a^2 - b^2[/latex]:
  • Step 1: Recognize the pattern [latex]a^2 - b^2 = (a+b)(a-b)[/latex].
  • Step 2: Substitute back the values of [latex]a[/latex] and [latex]b[/latex] to factorize.

5. Factoring Perfect Square Trinomials

• For trinomials like [latex]a^2+2ab+b^2[/latex]:
  • Step 1: Identify the square roots of the first and last terms.
  • Step 2: Ensure the middle term is twice the product of these roots, then factor as [latex](a+b)^2[/latex] or [latex](a-b)^2[/latex].

6. Factoring Cubes

• For expressions like [latex]a^3+b^3[/latex] or [latex]a^3 - b^3[/latex]:
  • Apply the sum or difference of cubes formula: [latex](a^3+b^3) = (a+b)(a^2+ab+b^2)[/latex] and [latex]a^3-b^3 = (a-b)(a^2+ab+b^2)[/latex]