Problem 78 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(2(x - 3y)\).

Step 1: Factor the Numerator

The numerator is \(8x - 24y\). To factor it, find the greatest common factor (GCF) of the terms. The GCF of 8 and 24 is 8. Factor out 8 from each term: \(8(x - 3y)\).

Step 2: Rewrite the Fraction

Replace the numerator \(8x - 24y\) with its factored form. The fraction then becomes: \(\frac{8(x - 3y)}{4}\).

Step 3: Simplify the Fraction

To simplify, look for common factors in the numerator and the denominator. The common factor is 4 (since \(8=2\times4\)). Divide both the numerator and the denominator by 4: \(\frac{8 \div 4 (x - 3y)}{4 \div 4}\) which simplifies to \(\frac{2(x - 3y)}{1}\).

Step 4: Simplify Further if Possible

Since \(\frac{2(x - 3y)}{1} = 2(x - 3y)\), and it is already simplified since there are no further common factors or expressions to reduce.

Key Concepts

Factoring Concepts

Factoring is the process of breaking down an expression into simpler "factors" that multiply together to form the original expression. Consider the expression \(8x - 24y\). The goal is to express this as a simpler product. This involves identifying common elements that can be "factored out" from each term. For example, in \(8x - 24y\), the term "8" is common because both 8 and 24 can be divided by 8 evenly. Factoring it out results in \(8(x - 3y)\). This simply means that the multiplication of 8 by the expression \((x - 3y)\) will yield the original numeric and variable terms in the expression.

• Factoring simplifies expressions, making other operations more manageable.
• Always look for the greatest common element to factor out.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest number that divides two or more numbers evenly. Identifying the GCF is crucial in factoring because it helps reduce things to simpler terms. For example, when factoring \(8x - 24y\), we first look for numbers that evenly divide both 8 and 24. The number 8 is the largest number that does this, making it the GCF. By factoring out 8, we create a simpler expression, \(8(x - 3y)\). Finding the GCF involves:

• Listing all divisors of each number.
• Identifying the largest common divisor.

Utilizing the GCF reduces expressions both in complexity and size, allowing for more straightforward mathematical operations going forward.

Simplification

Simplification is the process of refining a mathematical expression to its simplest form. After factoring, simplification may involve canceling out terms in a fraction to reduce it further. Take the fraction \(\frac{8(x - 3y)}{4}\). In this expression, both the numerator and the denominator share a common factor of 4. That means we can divide each by 4, effectively simplifying the fraction. Simplification results in \(\frac{2(x - 3y)}{1}\), which is the same as just \(2(x - 3y)\). This process involves:

• Identifying common factors across the numerator and denominator.
• Reducing the expression by canceling the shared terms.

This technique is immensely useful, especially in algebraic and fractional operations, ensuring expressions remain easy to manage and understand.

Numerator and Denominator

In a fraction, the numerator is the top part and the denominator is the bottom. Understanding their roles is crucial when working with fractions. For the expression \(\frac{8x - 24y}{4}\),

• The numerator is \(8x - 24y\), which includes both numerical and variable terms.
• The denominator is simply the number 4.

Factoring applies primarily to the numerator in this context because it contains the variables which can be manipulated further. Once the numerator is factored and simplified, we can then look for common factors between it and the denominator to further simplify the entire fraction. The numerator contains the heart of the algebraic expression, while the denominator primarily serves to define the division and scale of the fraction. This interplay is a fundamental aspect of understanding how fractions operate in algebraic expressions.