Problem 78 - Algebra A Combined Function

Question

Factor each numerator and denominator. Then simplify if possible. $$ \frac{8 x-24 y}{4} $$

Step-by-Step Solution

Answer

The expression simplifies to \(2(x - 3y)\).

Step 1: Factor the Numerator

First, observe the numerator, which is \(8x - 24y\). We can factor out the greatest common factor (GCF) of the terms 8 and 24. The GCF is 8. Factoring out 8 gives us \(8(x - 3y)\).

Step 2: Simplify the Fraction

Next, observe the entire fraction \(\frac{8(x - 3y)}{4}\). Since we have a common factor of 8 in the numerator and a factor of 4 in the denominator, we can simplify the fraction by dividing both numerator and denominator by 4. This gives us \(\frac{8}{4} \times (x - 3y)\).

Step 3: Reduce the Common Factor

After dividing, \(\frac{8}{4}\) simplifies to 2. Therefore, the fraction simplifies to \(2(x - 3y)\).

Key Concepts

Factoring

Factoring is the process of breaking down an expression into a product of simpler expressions, known as factors. Think of it as finding what you can multiply to get the original number or expression. When dealing with algebraic expressions, we often look for the greatest factor that each term has in common. This allows us to simplify expressions more easily later on. For example, in the expression \(8x - 24y\), we note that both terms can be divided by 8. By factoring out this greatest common factor, we rewrite the expression as \(8(x - 3y)\). Once factored, the expression is much cleaner and easier to work with in further calculations.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest number or expression that divides two or more terms without leaving a remainder. It is a crucial step in the simplification of algebraic fractions. To find the GCF between two terms like 8 and 24, consider both the numerical coefficients and, if applicable, the variables they share:

• Identify the largest number that divides into both coefficients.
• In this example, the GCF of the numbers 8 and 24 is 8 because both are divisible by 8.
• If variables are involved, choose the lowest power of each common variable.

This factor is then used to simplify expressions, as it allows us to factor expressions and subsequently reduce fractions.

Fraction Simplification

Fraction simplification involves reducing a fraction to its simplest form, where the numerator and denominator share no common factors other than 1. After factoring, the numerator and denominator can often be reduced by dividing each by their greatest common factor. In the fraction \(\frac{8x - 24y}{4}\), once we've factored the numerator to \(8(x - 3y)\), look to simplify it:

• Divide both the numerator and the denominator by the GCF of 4.
• This simplifies \(\frac{8}{4} \times (x - 3y)\) to \(2(x - 3y)\).

By performing these steps, you ensure the expression is as concise as possible, making it more interpretable and ready for further use in problem-solving.