In Exercises 1–68, factor completely, or state that the polynomial is prime. 12x³y − 12xy³

Verified step by step guidance

1. Identify the greatest common factor (GCF) of the terms in the polynomial.
2. The GCF of the terms 12x^3y and -12xy^3 is 12xy.
3. Factor out the GCF from each term in the polynomial.
4. Rewrite the polynomial as 12xy(x^2 - y^2).
5. Recognize that x^2 - y^2 is a difference of squares, which can be factored further as (x - y)(x + y).

Key Concepts

Factoring Polynomials

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.

Greatest Common Factor (GCF)

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.

Polynomial Degree and Terms

The degree of a polynomial is the highest power of the variable in the expression, which determines its behavior and the number of roots it can have. Understanding the terms of a polynomial, which are the individual components separated by addition or subtraction, is essential for effective factoring, as it helps in identifying patterns and applying appropriate factoring techniques.