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When tackling problems like factoring polynomials, identifying the greatest common factor (GCF) is an essential first step. The GCF is the largest factor that all terms in an algebraic expression share. In other words, it's the biggest chunk you can pull out of every term
without leaving a remainder.To find the GCF, you'll want to list all the factors of each term. For instance, in our expression \(42y^2 - 6y\), we need to look at the numbers \(42\) and \(6\), as well as the common variable \(y\).

• The greatest common factor of \(42\) and \(6\) is \(6\), since both are divisible by \(6\).
• Each term also contains a \(y\), making the GCF \(6y\).

Recognizing the GCF as \(6y\) helps simplify the process of factoring and allows you to rewrite the expression in a more manageable form.

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition or subtraction. Understanding how to manipulate these expressions is critical in algebra.
They represent quantities that can vary or depend on one or more variables.In our example, \(42y^2 - 6y\) is an algebraic expression. Here, \(y\) is the variable, \(42\) and \(6\) are coefficients, and the operation between the terms is subtraction. Working with algebraic expressions involves recognizing patterns and using rules to simplify or factor them.
Knowing the structure allows us to effectively apply mathematical processes like finding the GCF or using the distributive property. Recognizing these elements within an expression can make it easier to solve or factor, leading to accurate solutions.

The distributive property is a fundamental algebraic principle used to simplify expressions and solve equations. It allows us to multiply a single term across a binomial (or more complex expression) and is key to both expanding and factoring expressions.In our problem, once we identified the GCF \(6y\), the distributive property was used in reverse to factor the expression as \(6y(7y - 1)\). This demonstrates its use in factoring, rewriting a product as a simpler expression.

• Factoring: The distributive property can transform an expanded form like \(42y^2 - 6y\) into a factored form \(6y(7y - 1)\).
• Checking Work: After factoring, we distributed \(6y\) back across \(7y - 1\) to ensure we returned to the original expression, validating our factorization.

Using the distributive property effectively means understanding how to both break down and reassemble expressions, a crucial skill in algebra.