Factor by pulling out the greatest common factor first, then apply patterns such as difference of squares or trinomial factoring. Verify by multiplying factors back to the original polynomial.
- Always check for a common factor before anything else.
- Pattern recognition speeds factoring but still requires verification.
- Refactoring after expansion is the best correctness check.
Apply a repeatable factoring order
A reliable order is: factor out GCF, look for special products, then factor remaining trinomials or use grouping. This prevents missed factors and partial answers.
Difference of squares factors as a^2 - b^2 = (a - b)(a + b). Perfect square trinomials and sum or difference of cubes also have known patterns.
Factor trinomials with care
For x^2 + bx + c, find two numbers that multiply to c and add to b. For ax^2 + bx + c with a not equal to 1, methods such as the AC approach or decomposition by grouping are common.
- Track signs using product and sum logic.
- Write candidate factor pairs before selecting one.
- Multiply back to verify every term.
Factor a quadratic with leading coefficient
A geometry expression gives area polynomial 6x^2 + 11x + 3.
- Compute a*c = 6*3 = 18 and find pair summing to 11: 9 and 2.
- Split middle term: 6x^2 + 9x + 2x + 3.
- Group: 3x(2x + 3) + 1(2x + 3).
- Factor common binomial: (3x + 1)(2x + 3).
Common mistakes
- Skipping GCF and leaving factors incomplete.
- Using factor pairs that match product but not sum.
- Losing sign accuracy when splitting middle terms.
- Not multiplying factors back for confirmation.
Try one
Factor x^2 - 16.
It is a difference of squares: (x - 4)(x + 4).
Sources
- OpenStax: College AlgebraOpen textbook with full chapter coverage, worked examples, and practice problems.
- Khan Academy: Algebra and Algebra 2Official lessons and practice sets for core algebra skills.