Question

Check all of the possible first steps in factoring a polynomial with four terms.

factor out a GCF

factor the difference of cubes

factor a sum of cubes

factor a difference of squares

factor a perfect-square trinomial

factor by grouping

Answer 1

Possible first steps for factoring a polynomial with four terms:

- ✓ factor out a GCF — Yes. Always check and factor out a greatest common factor first.
- ✗ factor the difference of cubes — No as a direct first step; this pattern applies to a 3-term expression of the form a^3 − b^3, so you would only use it after reducing the expression (e.g., after grouping).
- ✗ factor a sum of cubes — No as a direct first step for the same reason (applies to a^3 + b^3).
- ✗ factor a difference of squares — Not usually a direct first step for a 4-term polynomial (it applies to two-term expressions); it can be used later after grouping or simplifying.
- ✗ factor a perfect-square trinomial — Not a direct first step because it applies to trinomials; it can be used later if grouping produces a suitable trinomial.
- ✓ factor by grouping — Yes. Grouping is the standard method for four-term polynomials (often after factoring out a GCF).

In short: always try factoring out a GCF and then try factoring by grouping; the other patterns may appear later after those steps.