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Factoring common factors is a crucial step when dealing with rational expressions. It involves identifying and extracting the greatest common factor (GCF) in an expression, which can simplify calculations later on. In the given exercise, the numerator of the expression is \( 30(f+2) \). Here, \( 30 \) is factored out as the common multiplier.

• Identify the GCF in the terms of the expression. Look at both numbers and variables.
• Write the expression as a product of its GCF and the remaining terms.

By taking out \( 30 \), we are left with an expression that looks simpler and is easier to manage in subsequent steps. When terms have common factors, like \((f+2)\) within the numerator, factoring also allows for potential simplification with the denominator.

Simplifying expressions, particularly rational ones, means reducing them to their simplest form. The purpose of simplification is to make the expression as straightforward as possible. This involves canceling out terms and reducing coefficients.

Cancel out common terms

Look for terms that appear in both the numerator and the denominator. If such common terms exist, they can be canceled. In our example, \((f+2)\) appears in both parts of the fraction and can be eliminated from the equation.

Reduce coefficients

Once you've canceled common terms, check the numerical coefficients in your expression.

• Divide both the numerator and the denominator by their GCF.
• In our problem, dividing \( 30 \) by \( 6 \) simplifies it to \( 5 \).

Simplification ensures that equations are in their most manageable form, making it easier to understand and use in further calculations.

Writing rational expressions in lowest terms is the process of simplifying them as much as possible, so they represent the same value but with the fewest factors. This is essential when solving math problems, as it reduces the complexity of the expression.To express a fraction in its lowest terms:

• Factor both the numerator and the denominator completely.
• Cancel out all common factors.
• Ensure no other common factors exist between the new numerator and denominator.

In this exercise, after canceling out \((f+2)\) and dividing through by the GCF, \( 6 \), the expression \( \frac{5}{(f-5)} \) is the simplest version of the original fraction. It is now in its lowest terms, which means that it can't be simplified any further, and no common factors remain.