91Ó°ÊÓ

To begin with, let's delve into the concept of a common factor. In algebra, a common factor is an expression that is a part of two or more terms. Identifying a common factor is like finding what numbers or variables are shared among terms in an algebraic expression. For example, in the given practice problem, we look at the terms \( xy \) and \( -5y \) and notice that \( y \) appears in both. Similarly, \( -2x \) and \( 10 \) share the number \( 2 \) (though \( 10 \) is technically \( -2 \times -5 \) which makes \( -2 \) the common factor).

When factoring polynomials that are not immediately obvious to simplify, the grouping method is an efficient strategy. The process includes several steps:

• First, divide the expression into groups where a common factor can be identified.
• Next, factor out any common factors from each group.
• Lastly, if possible, factor out from the resulting expression.

In the provided exercise, we grouped the terms into \( (xy - 5y) \) and \( -(2x - 10) \) by rearranging them, which allowed us to recognize \( y \) as a common factor in the first group and \( -2 \) in the second, paving the way for further simplification.

An algebraic expression, such as the one in our example \( xy-5y-2x+10 \), is a combination of constants, variables and arithmetic operations. Factoring algebraic expressions involves rewriting these expressions as a product of simpler or more revealing terms, which can greatly simplify solving equations or evaluating the expressions. The final step in our exercise was to recognize that \( x - 5 \) was a common factor in the result of our grouping efforts. By factoring \( x - 5 \) out of the grouped terms, we were left with the much simpler expression \( (y - 2)(x - 5) \), completing the factoring process and revealing the underlying relationship between the variables in the original algebraic expression.