Problem 79 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{2 x^{2}-3 x-9}{2 x^{2}+3 x-9} $$

Step-by-Step Solution

Answer

Does not simplify.

Step 1: Identify the Expression Components

First, identify the numerator and the denominator of the expression. The numerator is \( 2x^2 - 3x - 9 \) and the denominator is \( 2x^2 + 3x - 9 \).

Step 2: Look for Common Factors

Check if there are any common factors between the numerator and the denominator. In this case, there are no common factors that can be easily seen.

Step 3: Attempt Factoring the Numerator

Try to factor the numerator \( 2x^2 - 3x - 9 \). Look for two numbers that multiply to \( 2 imes -9 = -18 \) and add to \(-3\). This can be factored as \( (2x + 3)(x - 3) \).

Step 4: Attempt Factoring the Denominator

Try to factor the denominator \( 2x^2 + 3x - 9 \). Look for two numbers that multiply to \( 2 imes -9 = -18 \) and add to \(3\). This can be factored as \( (2x - 3)(x + 3) \).

Step 5: Check for Common Factors Post-Factoring

After factoring both the numerator and the denominator, check if there are common factors. The numerator is \( (2x + 3)(x - 3) \) and the denominator is \( (2x - 3)(x + 3) \), and they do not share any common factors.

Step 6: Determine Simplification Possibility

Since there are no common factors between the factored numerator and the factored denominator, the expression cannot be simplified further. This means the given expression does not simplify.

Key Concepts

Factoring

Factoring is a crucial step when dealing with rational expressions. It involves breaking down a polynomial into a product of simpler polynomials, which can help identify opportunities for simplification. To factor a quadratic expression like the ones in our example, you'll first want to find two numbers that multiply to give the constant term's multiple with the leading coefficient (in other words, the product of the coefficient of the squared term and the constant term) and add up to the middle term's coefficient.
This approach is effectively applied:

• Numerator: For the expression \(2x^2 - 3x - 9\), look for two numbers that multiply to \(-18\) and add to \(-3\). The numbers \(3\) and \(-6\) satisfy these conditions, allowing us to factor it as \((2x + 3)(x - 3)\).
• Denominator: For the expression \(2x^2 + 3x - 9\), we look for numbers that multiply to \(-18\) and add to \(3\), which are \(6\) and \(-3\). This results in the factorization \((2x - 3)(x + 3)\).

Factoring reveals hidden commonalities and is a fundamental skill for manipulating and simplifying rational expressions.

Numerator and Denominator

Understanding the concepts of numerator and denominator is essential when working with fractions, including rational expressions.
The numerator is the top part of the fraction, while the denominator is the bottom part. In our example, the numerator is the expression \(2x^2 - 3x - 9\), and the denominator is \(2x^2 + 3x - 9\).
Here's why they matter:

• Identifying each part separately helps in organizing the expression, which is crucial before starting the process of simplification.
• The numerator and the denominator hold the key to evaluating or manipulating a rational expression by determining its value and simplifying its form.

In any simplification process, changes to these parts determine whether the expression simplifies further or not.

Common Factors

Common factors are identical factors that exist in both the numerator and the denominator of a rational expression. They are crucial for simplification as their presence allows you to "cancel" or reduce the expression.
Unfortunately, in the given exercise, after factoring both the numerator and the denominator, we find no common factors:

• Numerator after factoring: \((2x + 3)(x - 3)\)
• Denominator after factoring: \((2x - 3)(x + 3)\)

Because these expressions do not share any factors, the original rational expression cannot be simplified further.
Common factors simplify rational expressions by reducing them to their simplest form, making them easier to work with or solve.

Simplification of Expressions

Simplification of rational expressions involves reducing them to their simplest form by canceling out common factors in the numerator and denominator.
This process typically follows these steps:

• Separate the expression into the numerator and the denominator.
• Factor both parts completely.
• Identify and cancel any common factors.

In our exercise, although both the numerator \((2x^2 - 3x - 9)\) and the denominator \((2x^2 + 3x - 9)\) were successfully factored, no common factors were identified. Thus, the expression does not simplify further. This highlights an important point: not all rational expressions can be simplified.
Understanding the entire simplification process allows us to pinpoint when and why a rational expression may become simpler or remain unchanged.