General Strategy for Factoring Polynomials
To completely factor a polynomial, follow a systematic three-step general strategy.
Step 1. Check for a Greatest Common Factor (GCF) among all terms and factor it out if one exists.
Step 2. Classify the polynomial by its number of terms to determine the appropriate factoring method:
- Binomials (two terms): Check for special patterns. Is it a difference of squares? If so, factor it as the product of conjugates. Is it a sum or difference of cubes? Use the corresponding cube pattern. Note that a sum of squares cannot be factored over the real numbers.
- Trinomials (three terms): If it has the form , "undo FOIL" to find the binomial factors. If it has the form , first check if it fits the perfect square trinomial pattern where the first and last terms are perfect squares. If it does not, apply the trial and error method or the "ac" method.
- Polynomials with more than three terms: Use the factoring by grouping method.
Step 3. Check the result. Verify that the polynomial is factored completely (meaning no factors can be broken down further) and multiply the factors together to ensure they produce the original polynomial.
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