Problem 33 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{-5 a-5 b}{a+b} $$

Step-by-Step Solution

Answer

The simplified expression is \(-5\).

Step 1: Factor the Numerator

First, notice that the numerator \(-5a - 5b\) has a common factor of \(-5\). Factoring the common factor out, we get \(-5(a + b)\).

Step 2: Examine the Expression

The expression is now \(\frac{-5(a + b)}{a + b}\). Here, the term \(a + b\) appears in both the numerator and the denominator.

Step 3: Cancel Common Factors

Since \(a + b\) in the numerator and the denominator are the same, and assuming \(a + b eq 0\), we can cancel them out, resulting in \(-5\).

Step 4: Write the Simplified Expression

After canceling the common terms, the simplified expression is just \(-5\).

Key Concepts

Factoring in Algebra

Factoring in algebra involves breaking down expressions into simpler pieces that multiply together to create the original expression. It makes complex expressions easier to work with by revealing the common components. In the expression \(-5a - 5b\), we can see that each term has a common factor of \-5\. Here's a simple approach to factor:

• Identify terms that can be grouped with a common factor.
• In \(-5a - 5b\), both terms are divisible by \-5\.
• Factor out the common factor, which gives \(-5(a + b)\).

This not only simplifies your work but also helps spot common factors in fractions.

Canceling Common Factors

Once you have factored an expression, the next step is to check for common terms in both the numerator and the denominator of a fraction. Canceling common factors simplifies the fraction further and makes calculations easier. Here's how it works in our example:

• Start with the expression \(\frac{-5(a + b)}{a + b}\).
• Notice that \(a + b\) appears both at the top and bottom.
• As long as \(a + b eq 0\), these terms can cancel out.

After cancelation, what remains is just \(-5\). Such simplification is vital in algebra because it can turn complicated fractions into manageable ones.

Algebraic Fractions

An algebraic fraction consists of numerators and denominators that are algebraic expressions—polynomials, in most cases. Understanding how to simplify these fractions is key to dealing with more involved algebraic problems. Here's a brief rundown of how to handle these simplifications:

• Start by factoring both the numerator and the denominator as much as possible.
• Look for any common factors that appear in both parts of the fraction.
• Cancel these factors to simplify the fraction entirely.

When the fractions are simplified, it makes solving equations or further operations much more straightforward. In our example, simplifying the expression and canceling \(a + b\) results in the simplified value \(-5\). Mastery of these techniques can save time and reduce errors.