Factoring & polynomial identities
The idea
Factoring runs multiplication in reverse: it rewrites a polynomial as a product of simpler pieces, the way 60 = 2² × 3 × 5 breaks an integer into primes. Building on your work with expressions, exponents, and integer factors, the skill is recognizing structure: a shared common factor, the difference of squares a² − b² = (a − b)(a + b), perfect-square trinomials, and grouping patterns. Factored form is powerful because a product is zero exactly when one of its factors is zero — so factoring turns equation solving into reading off roots.
Work in a fixed order: pull out the greatest common factor first, then look for a recognizable identity, then try grouping on four-term polynomials. After each move, ask whether any factor can still be broken down; factoring completely means no factor hides a remaining pattern. A common misconception is treating a² + b², the sum of squares, like the difference of squares — over the real numbers it does not factor, and forcing (a + b)(a + b) onto it invents a middle term that does not exist. Multiply your factors back together when in doubt; factoring claims are cheap to verify.
Worked example
Factor 2x³ + 3x² − 8x − 12 completely.
- No factor is shared by all four terms, but four terms suggest grouping. Pair them: (2x³ + 3x²) + (−8x − 12).
- Factor each pair: x²(2x + 3) from the first and −4(2x + 3) from the second. Both groups now expose the same binomial, which is the signal that grouping is working.
- Pull out the common binomial: x²(2x + 3) − 4(2x + 3) = (2x + 3)(x² − 4).
- The second factor is a difference of squares, x² − 4 = (x − 2)(x + 2), so the complete factorization is (2x + 3)(x − 2)(x + 2).
- Check with x = 1: the original gives 2 + 3 − 8 − 12 = −15, and the factors give (5)(−1)(3) = −15. The values agree, supporting the factorization.
Answer. 2x³ + 3x² − 8x − 12 = (2x + 3)(x − 2)(x + 2).
Check your understanding
- Why does the zero-product property make factored form so much more useful than expanded form for solving equations?
- How do you decide which factoring strategy to try first, and what clues in a polynomial point toward grouping?
- Why does x² + 4 refuse to factor over the real numbers while x² − 4 splits so easily?
- How would you convince a skeptic that your factorization is both complete and correct without re-deriving it from scratch?
Build the foundations first
Factoring & polynomial identities builds on these concepts. If any feel shaky, start there.