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When dealing with algebraic expressions, understanding the concept of like terms is crucial. Like terms are terms in an expression that have the same variables raised to the same powers. This means that their variable part is identical, even if their coefficients are different. Recognizing like terms allows you to simplify expressions by combining them.

For instance, in the expression \((2x)(3y) + 4x + 5y + (6x)(3y)\), the terms \((2x)(3y)\) and \((6x)(3y)\) are like terms. Why? Because they both contain the product of the variables \(x\) and \(y\). The coefficient may differ, but because the variable factors are the same, they are classified as like terms.

Remember, only terms with identical variable parts can be combined. Thus, \(4x\) and \(5y\) are not like terms to any of the others in this expression, as they contain, respectively, only \(x\) and only \(y\) without the other factor.

After identifying like terms, the next step is to combine them. This process involves performing operations on the coefficients of the terms while maintaining the shared variable part. In the expression given, \((2x)(3y) + (6x)(3y)\) turns into \(6xy + 18xy\) when the internal multiplications are performed:

• \((2x)(3y) = 6xy\)
• \((6x)(3y) = 18xy\)

Now, with these results, you can combine them because they are like terms.

Combining means adding or subtracting the coefficients while keeping the common variables untouched, resulting in \(6xy + 18xy = 24xy\). This combines the terms into a single simplified term.

The ability to combine like terms simplifies expressions significantly, reducing them to a form that is easier to understand and work with.

Polynomial simplification involves reducing an expression to its simplest form. This process uses the identification and combination of like terms to condense complex expressions. The example expression is simplified in this way:

• First, identify and perform any necessary operations on individual terms to express the expression in its raw components, as seen with \((2x)(3y)\) and \((6x)(3y)\) resulting in \(6xy\) and \(18xy\).
• Combine these like terms to simplify: \(6xy + 18xy = 24xy\).

Finally, write down the new expression, adding any remaining non-like terms at the end to reach the final simplified form: \(24xy + 4x + 5y\).

This process not only makes expressions easier to read and evaluate but also lays the groundwork for further algebraic operations. Simplified expressions often reveal patterns and relationships not immediately obvious in their extended forms.