In Exercises 1–68, factor completely, or state that the polynomial is prime. 12x³ + 3xy²

Factoring polynomials involves breaking down a polynomial expression into simpler components, or factors, that when multiplied together yield the original polynomial. This process often includes identifying common factors, applying the distributive property, and recognizing special polynomial forms such as the difference of squares or perfect square trinomials.

The greatest common factor (GCF) is the largest factor that divides two or more numbers or terms without leaving a remainder. In polynomial expressions, finding the GCF is crucial as it simplifies the factoring process by allowing you to factor out the GCF from each term, making the remaining polynomial easier to work with.

A prime polynomial is a polynomial that cannot be factored into simpler polynomials with integer coefficients. Recognizing a polynomial as prime is essential when factoring, as it indicates that the polynomial does not have any factors other than itself and one, thus concluding the factoring process.