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In algebra, finding the Greatest Common Factor (GCF) is a crucial step for simplifying expressions and factoring polynomials. To find the GCF of a polynomial, identify the largest expression that divides each term without leaving a remainder.

In our example, the expression is \(x^4 y^3 - x^2 y^5\). Each term consists of variables \(x\) and \(y\), which can be examined individually. For \(x\), the lowest power is \(x^2\), and for \(y\), it's \(y^3\). Hence, \(x^2 y^3\) is the GCF.

By pulling out \(x^2 y^3\), you simplify the original expression into a smaller and more manageable factor:

• Breaking down terms: \(x^4 y^3\) becomes \(x^2 y^3 \cdot x^2\), and \(-x^2 y^5\) becomes \(x^2 y^3 \cdot (-y^2)\).
• Resulting in: \(x^2 y^3(x^2 - y^2)\).

Finding the GCF streamlines the process and prepares the expression for deeper factoring steps.

Factoring polynomials often involves recognizing specific patterns, such as the "difference of squares." This pattern arises when you have two squared terms separated by a minus sign.

In the factored form \(x^2 y^3(x^2 - y^2)\), the expression \(x^2 - y^2\) presents this pattern:

• The expression can be rewritten as a product of two binomials: \((x + y)(x - y)\).
• This draws from the identity: \(a^2 - b^2 = (a + b)(a - b)\).

Recognizing the "difference of squares" lets us break down expressions into simpler binomial factors, facilitating easier computation and understanding.
Remember, this technique only works when both terms are perfect squares and they're subtracted.

Complete factorization is the art of breaking down a polynomial into its simplest building blocks or factors. By working through the expression step-by-step, you can achieve full factorization, which unlocks deeper insights into the behavior of the polynomial.

Starting with the expression \(x^4 y^3 - x^2 y^5\), we found the GCF as \(x^2 y^3\), enabling us to express it as \(x^2 y^3 (x^2 - y^2)\). Recognizing \(x^2 - y^2\) as a difference of squares further allows factoring into \((x+y)(x-y)\).

Complete factorization for our expression is:

• \(x^2 y^3 (x + y)(x - y)\)

Each factor cannot be divided further using integers or the original variables, ensuring it's fully simplified. Completing factorization simplifies expression evaluation, solving, and comparison.