In Exercises 1–68, factor completely, or state that the polynomial is prime. x⁶y⁶ − x³y³

Verified step by step guidance

1. Factor out the greatest common factor (GCF) from the polynomial. The GCF of the terms \(x^6y^6\) and \(x^3y^3\) is \(x^3y^3\).
2. Write the polynomial as \(x^3y^3(x^3y^3 - 1)\).
3. Recognize that \(x^3y^3 - 1\) is a difference of cubes.
4. Use the difference of cubes formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), where \(a = x^3y^3\) and \(b = 1\).
5. Apply the formula to factor \(x^3y^3 - 1\) as \((x^3y^3 - 1)((x^3y^3)^2 + x^3y^3 \cdot 1 + 1^2)\).

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 divides two or more numbers or expressions without leaving a remainder. In polynomial expressions, identifying the GCF allows for the simplification of the polynomial by factoring it out. This step is crucial in the factoring process, as it can reveal further factors and simplify the expression significantly.

Exponents and Power Rules

Exponents represent repeated multiplication of a base number, and understanding the rules of exponents is vital for manipulating polynomial expressions. Key rules include the product of powers, power of a power, and the difference of powers. In the given polynomial, recognizing how to combine and simplify terms with exponents is essential for effective factoring.