91Ó°ÊÓ

Factoring is a crucial step whenever you're working with algebraic expressions. It's all about identifying and extracting common factors amongst terms. In the given exercise, we focus on the numerator, \(-5a - 5b\). Notice the presence of \(-5\) in both terms. This is the greatest common factor (GCF) here. To factor it out, you simply divide each term in the expression by \(-5\), and group them together as a product.
Here's how it looks:- Start by splitting each term in the numerator: - \(-5a = -5 \times a\) - \(-5b = -5 \times b\)- Extract \(-5\) and rewrite the expression as: \(-5(a + b)\).Factoring simplifies the expression considerably, making it easier to identify further simplifications, such as those involving identical terms in both the numerator and the denominator.

Canceling is the process of removing common factors or identical terms from the numerator and the denominator. It leads to simplification of fractions, whether algebraic or numerical. In our example, upon factoring the numerator, we have \(-5(a + b)\). The denominator is \(a + b\), which matches the term in the numerator.
To cancel, remember:- Terms must be identical in both numerator and denominator.- They must not be zero, so the expression remains valid.For this exercise:- Observe that \(a + b\) appears in both the numerator and the denominator.- Assuming \(a + b eq 0\), you can cross them out.Canceling reduces the complexity of the expression, leaving you with \(-5\), which is much simpler and compact.

Algebraic fractions can initially appear intimidating, as they resemble fractions filled with variables and expressions rather than plain numbers. Essentially, they function under the same basic rules as numerical fractions. You can simplify them by:

• Factoring numerators and denominators
• Canceling identical terms

For instance, in the expression \(\frac{-5a-5b}{a+b}\), after factoring and canceling the common terms \((a+b)\), you simplify it to \(-5\). This simplification process helps make complex expressions more manageable.
Remember that handling algebraic fractions with care is crucial to avoid division by zero errors, especially when canceling terms. Always verify the original expression allows for the cancellation of terms by ensuring the values of the variables maintain validity across each step.