Problem 32 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{3 x-9}{4 x-16} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{3(x-3)}{4(x-4)}\).

Step 1: Identify Common Factors

First, identify any common factors in the numerator and denominator. The numerator is \(3x-9\), which can be factored as \(3(x-3)\). The denominator is \(4x-16\), which can be factored as \(4(x-4)\).

Step 2: Factor Numerator and Denominator

Rewrite the expression using the factored forms from Step 1. The expression becomes \(\frac{3(x-3)}{4(x-4)}\).

Step 3: Cancel Common Factors

Check if there are any common factors in the numerator and denominator that can be canceled. In this case, there are no common factors to cancel, so the expression in its simplified form remains \(\frac{3(x-3)}{4(x-4)}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is an important skill that helps us simplify rational expressions effectively. The primary goal is to break down a polynomial into simpler parts, called factors, that when multiplied give back the original polynomial. This makes it much easier to spot common elements between different expressions.

When factoring, start by looking for the greatest common factor (GCF) of the terms in the polynomial. For the numerator in our exercise, which is \(3x - 9\), we notice that both terms share a factor of 3. Thus, we can factor the expression into \(3(x-3)\). Similarly, for the denominator \(4x - 16\), we observe that both terms have a common factor of 4, allowing us to rewrite it as \(4(x-4)\).

Factoring allows us to simplify complex expressions by breaking them down into more manageable components. This is a foundational concept in algebra that is used repeatedly in various branches of mathematics.

Common Factors

Identifying and understanding common factors is essential when simplifying rational expressions. A common factor of two or more numbers, or algebraic expressions, is a factor that divides each of them evenly.

To find common factors, look for numbers or expressions that evenly divide both the numerator and the denominator. These factors can often be extracted using the greatest common factor (GCF) method. In our exercise, after factoring, we end up with the terms \(3(x-3)\) in the numerator and \(4(x-4)\) in the denominator. Here, the process of searching for common factors would have us look for any brackets or numerical coefficients that match in both the numerator and the denominator.

In the simpler example of fractions, like \(\frac{6}{8}\), the common factor is 2, which simplifies the fraction to \(\frac{3}{4}\). The same principle applies to algebraic expressions, leading us to cancel matching terms when they exist.

Canceling Terms

Canceling terms is the final step in simplifying rational expressions. After factoring both the numerator and the denominator, you look for any factors that appear in both expressions. These are the common terms you can cancel.

Once you find these common terms, you can cancel them out by dividing both the numerator and the denominator by these terms, effectively simplifying the expression. In our exercise, the factored forms \(3(x-3)\) and \(4(x-4)\) had no terms common between them, so no further cancellation was possible.

This step ensures that the rational expression is in its simplest form. By focusing on canceling terms only when they match perfectly, you maintain the integrity of the expression while making it easier to work with in future calculations. Remember, only identical terms can be canceled between the numerator and the denominator. Always double-check for possible errors that might arise during this step.