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When working with algebraic expressions, the first step in simplifying or factoring them is often to identify common factors. A common factor is a term that is present in each part of the expression. For instance, in the expression \(x^2 - 6x\), we can identify that both terms share a common factor: the variable \(x\).

• \(x^2\) has the factor \(x\).
• \(-6x\) also has the factor \(x\).

This commonality allows us to simplify the expression by factoring out the \(x\).

Identifying common factors is crucial because it simplifies the expression, making it easier to manage and solve. The process not only reduces the complexity but also prepares the expression for further factorization if needed. Always look for common numerical and variable parts in each term of the expression.

After identifying the common factors in an expression, the next step is to use an appropriate factoring technique to simplify it. For the expression \(x^2 - 6x\), we notice that \(x\) is the common factor. To factor this expression, we perform the following steps:

• Take the common factor \(x\) out of each term.
• Write it outside a set of parentheses.
• Inside the parentheses, write the remaining parts of each term.

So, factoring \(x^2 - 6x\) yields:

\ x(x - 6)\

This technique is called **factoring out** or **factoring by grouping**. It simplifies the expression and can also help in solving equations where the expression is set to zero. It also helps in recognizing patterns within the expression, enabling further factorization if possible.

Verification of factorization is a critical step that ensures the factored form of an expression is correct. Once an expression has been factored, you should always check your work. For our given expression \(x^2 - 6x\), which we factored as \(x(x - 6)\), we verify by distributing the factored terms back together:

\ x(x - 6) = x^2 - 6x\

Since the simplified expression matches the original, the factorization is correct.

Verification involves reversing the factoring process, often referred to as expanding. It guarantees that your work is accurate and that no terms were lost or altered in the factorization process. Therefore, always include this step to ensure the integrity of your solution. This not only builds confidence but also reinforces understanding of the relationship between distributed and factored forms. Reach for confirmation every time you factor an expression, just to be sure!