In Exercises 1–68, factor completely, or state that the polynomial is prime. x³y − 16xy³

Verified step by step guidance

1. Identify the greatest common factor (GCF) of the terms in the polynomial. In this case, the GCF is \( xy \).
2. Factor out the GCF \( xy \) from each term in the polynomial. This gives you \( xy(x^2 - 16y^2) \).
3. Notice that the expression inside the parentheses, \( x^2 - 16y^2 \), is a difference of squares.
4. Recall the difference of squares formula: \( a^2 - b^2 = (a - b)(a + b) \).
5. Apply the difference of squares formula to \( x^2 - 16y^2 \), where \( a = x \) and \( b = 4y \), to factor it further.

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 requires identifying common factors, applying special factoring techniques such as difference of squares, or using methods like grouping. Understanding how to factor is essential for simplifying expressions and solving polynomial equations.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest factor that two or more terms share. To factor a polynomial, identifying the GCF is often the first step, as it can simplify the expression significantly. For the polynomial x³y − 16xy³, recognizing that both terms share a common factor of xy allows for easier factoring and simplification.

Difference of Squares

The difference of squares is a specific factoring technique used when a polynomial can be expressed in the form a² - b², which factors into (a - b)(a + b). In the given polynomial, after factoring out the GCF, the remaining expression can be analyzed for patterns that fit this form, allowing for further simplification. Recognizing this pattern is crucial for complete factorization.