Recipe

Factor a polynomial completely

Use when: you're handed a polynomial and asked to write it as a product of factors, none of which can be split any further over the integers

Goal: the polynomial's complete factorization, with every factor checked and re-checked until nothing more comes apart

Taught in Polynomials and factoring

1. 1

   Scan every term and pull out the largest monomial, coefficient and variable part both, that divides all of them.

   Example

   6x^3 + 9x^2: both terms share 3x^2, so it factors as 3x^2(2x+3).

   ⚡ Shortcut

   If the leading coefficient is negative, pull out -1 along with the GCF so the leading term inside the parentheses turns positive. For example, -x^2-2x+15 factors far more reliably as -1(x^2+2x-15) = -(x+5)(x-3) than by hunting for p,q against a negative leading coefficient directly.

   Why: any GCF left sitting inside a factor makes every pattern below harder to spot, and it's the one step every other pattern silently assumes is already done.

2. 2

   However many terms remain after the GCF is out decides which pattern to try next.

   • 2 terms — go to Step 3 (a two-term pattern)
   • 3 terms — go to Step 4 (a trinomial)
   • 4 terms — go to Step 5 (grouping)

   Why: term count narrows a toolkit of five patterns down to the one or two that could possibly apply, before any guessing starts.

3. 3

   Check whether the two terms are both perfect squares with a minus between them (difference of squares), or both perfect cubes with a plus or minus between them (sum or difference of cubes). A sum of squares never factors over the reals, so don't go looking for one.

   Example

   x^2-4=(x-2)(x+2) (difference of squares). x^3-8=(x-2)(x^2+2x+4) (difference of cubes: same sign, opposite sign, always positive, nicknamed "SOAP").

   Why: these two shapes are recognized by pattern-matching, not by trial-and-error, so checking for them first saves the slower search in Step 4.

4. 4

   First check the perfect-square shortcut: are the first and last terms perfect squares, and does twice the product of their roots match the middle term? If so, it's (a\pm b)^2 instantly. Otherwise, find integers p,q with pq=c and p+q=b for x^2+bx+c. For a leading coefficient other than 1, find p,q with pq=ac and p+q=b instead, then split the middle term and factor by grouping.

   Example

   9x^2-30x+25: \sqrt{9x^2}=3x, \sqrt{25}=5, and 2(3x)(5)=30x matches the middle term, so it's (3x-5)^2 with no p,q-search needed.

   Why: the perfect-square check takes seconds and skips the whole p,q search whenever it hits. When it doesn't hit, p,q (or the ac-method) is the reliable fallback for every trinomial.

5. 5

   Pair the four terms (usually first-two, last-two), factor the GCF out of each pair separately, then factor out the shared binomial that remains.

   Example

   x^3+3x^2+2x+6 = x^2(x+3) + 2(x+3) = (x^2+2)(x+3).

   Why: four terms rarely share one common factor, but they often share a common binomial once each pair is factored on its own.

6. 6

   Look at each factor from Steps 3–5 and ask whether it has its own GCF, or is itself a difference of squares or another trinomial. If so, repeat the earlier steps on that factor.

   Why: "factored" doesn't mean "the first split you found": a factor that still splits further isn't factored completely yet.

7. 7

   Multiply every factor together and confirm the product matches the original polynomial.

   Why: this is the fastest way to catch a dropped sign or a mismatched factor pair before calling the answer final.