By Mara Pesacreta • Updated Aug 30, 2022

Polynomials are algebraic expressions that combine variables and constants using addition, subtraction, and exponents. Factoring simplifies these expressions by extracting common factors and applying algebraic identities.

Step 1: Identify the Polynomial Type

Determine whether the expression is a binomial (two terms) or a trinomial (three terms). Example binomial: 4x – 12. Example trinomial: x² + 6x + 9.

Step 2: Recognize Special Factoring Forms

Certain binomials follow patterns:

• Difference of squares: x² – y² = (x + y)(x – y)
• Difference of cubes: x³ – y³ = (x – y)(x² + xy + y²)
• Sum of cubes: x³ + y³ = (x + y)(x² – xy + y²)

Step 3: Extract the Greatest Common Factor (GCF)

Find the largest constant divisible by all coefficients. For 4x – 12, the GCF is 4:

4x – 12 = 4(x – 3)

Step 4: Factor Trinomials

For a trinomial ax² + bx + c, locate two numbers that multiply to ac and sum to b. Example:

Factor x² + 6x + 9: numbers 3 and 3 satisfy 3 × 3 = 9 and 3 + 3 = 6. Thus:

(x + 3)(x + 3)

Step 5: Verify Your Factorization

Multiply the factors back together to confirm you retrieve the original expression. Example:

4(x – 3) → 4x – 12 (matches the original).
(x + 3)(x + 3) → x² + 6x + 9 (matches the original).

Essential Tools

• Pen and paper
• Textbook or reliable online resource
• Calculator (for large numbers)