Problem 56 - Elementary Algebra

Question

For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms. \(\frac{8 x-16}{-4}\)

Step-by-Step Solution

Answer

Question: Simplify the given rational expression: \(\frac{8x-16}{-4}\) Answer: Simplify this expression by factoring the numerator and denominator, and cancelling out the common factors. The simplified expression is: \(-2(x - 2)\).

Step 1: Factor the numerator and the denominator

Factor the numerator and the denominator by recognizing the greatest common factor (GCF) in each expression. In this case, the GCF of \(8x-16\) is \(8\) and the constant \(-4\) in the denominator is just \(-4\). So, the expression can be factored as: $$ \frac{8x - 16}{-4} = \frac{8(x - 2)}{-4} $$

Step 2: Cancel common factors

Next, observe that both the numerator and the denominator have common factors which can be cancelled. In this case, the common factor is \(-4\). $$ \frac{8(x - 2)}{-4} = -2(x - 2) $$ So, the simplified expression is: $$ -2(x - 2) $$

Key Concepts

Greatest Common Factor

The Greatest Common Factor, often abbreviated as GCF, is fundamentally important in simplifying rational expressions. It represents the largest factor that divides two numbers or terms. For instance, when dealing with an expression like \(8x - 16\), we first identify the GCF of its terms. Here, both terms share a factor of \(8\), since \(8\) can divide both \(8x\) and \(16\) without leaving any remainder. By identifying the GCF, you can effectively "factor out" this number from the terms, simplifying the expression.

• Start by listing the factors of each term.
• Identify the largest factor common to both.
• The GCF is used to pull out or "factor" the commonalities in each term.

Understanding the GCF is crucial, as it lays the groundwork for both factoring and simplifying algebraic expressions. Without this step, you might overlook simplifications that could make solving the expression or equation much easier.

Factoring

Factoring is a process of expressing an algebraic expression as a product of its factors. When you "factor" something, you are essentially breaking it down into simpler components.
In the expression \(8x - 16\), once we identify \(8\) as the GCF, we can rewrite the expression as \(8(x - 2)\). This is factoring in action—rewriting the expression to show it as a product of its factors: the GCF (8) and the remaining term \((x - 2)\).

• Identify the GCF for the entire expression.
• Divide each term by the GCF to find the remaining terms inside the bracket.
• Reassemble into a factored form, which is often simpler to work with.

Factoring not only simplifies expressions but also helps in solving equations, as it makes the roots more obvious, allowing for easier further manipulation and simplification.

Simplification of Algebraic Expressions

Simplification of algebraic expressions involves reducing the expression to its simplest form. It ensures that the expression is as straightforward as possible, facilitating easier arithmetic operations or further algebraic manipulation.
After factoring an expression, the next step usually is simplification, where you can often cancel out common terms in the numerator and the denominator of a rational expression. In our example, \(\frac{8(x - 2)}{-4}\), the common terms we see are multiples of \(-4\) that can be cancelled. This process turns the expression into \(-2(x - 2)\), a much simpler form.

• Look for common factors in the numerator and the denominator.
• Cancel out these factors to reduce the expression.
• Ensure that the expression could not be further simplified.

Simplification is crucial not just for making calculations easier. It also aids in checking the correctness of equations and gauging the relationship between different mathematical expressions.