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The distributive property is a valuable tool in algebra that helps us rewrite expressions and factor polynomials efficiently. It essentially states that for any three numbers, a, b, and c, the equation \(a(b + c) = ab + ac\) holds true. This property is crucial when factoring polynomials because it allows us to "distribute" terms across others or "factor" a common term out.
For example, in the exercise provided, the expression \(x(2x - 5) - 3y(2x - 5)\) implements the distributive property. We notice that \((2x - 5)\) is a common factor in both terms. By applying the distributive property here, we can rewrite the expression as \((2x - 5)(x - 3y)\). This is because we're essentially saying that both initial terms can be multiplied by the same expression, \((2x - 5)\), consolidating them into a single product.
This property not only simplifies equations but also makes it easier to solve them or find their roots.

Finding common factors is a fundamental part of simplifying and factoring polynomials. A common factor is a term that appears in both parts of an expression. Identifying these can significantly simplify the process of factoring a polynomial.
In our original problem, we started with the expression \(2x^2 - 5x - 6xy + 15y\). To begin factoring this polynomial, we looked for common factors in the grouped terms:

• In the first pair, \(2x^2 - 5x\), the common factor is \(x\), giving us \(x(2x - 5)\).
• In the second pair, \(-6xy + 15y\), the common factor is \(y\), which simplifies to \(-y(6x - 15)\).

Identifying common factors is key to simplifying expressions and is usually the first step taken when trying to factor polynomials. It reduces complex polynomials into simpler expressions, making further operations more straightforward.

When factoring polynomials, grouping like terms is a strategy used to identify and extract common factors more easily. By rearranging the terms of a polynomial, it becomes simpler to see which terms can be grouped together based on common factors.
In the initial polynomial \(2x^2 - 5x - 6xy + 15y\), the terms were restructured to make the process of factoring more manageable. Let’s break it down:

• We looked at \(2x^2 - 5x\) and \(-6xy + 15y\), separating terms that could share a common factor.
• This grouping highlighted potential factors that each segment of the equation could share, such as \(x\) in the first group and \(y\) in the second.

This approach makes the process systematic, allowing step-by-step simplification of complex polynomials. This technique can be particularly useful when dealing with larger polynomials in algebra.