Simplifying (3x^2 - 12) / (10 - 5x) A Step-by-Step Guide
In mathematics, simplifying expressions is a fundamental skill. This article delves into the process of simplifying a specific rational expression: (3x^2 - 12) / (10 - 5x). We'll break down each step, explaining the underlying principles and techniques involved. Understanding how to simplify such expressions is crucial for various mathematical operations, including solving equations, graphing functions, and performing calculus.
Factoring: The Key to Simplification
The cornerstone of simplifying rational expressions lies in factoring. Factoring involves breaking down a polynomial into its constituent factors – expressions that, when multiplied together, yield the original polynomial. By identifying common factors in the numerator and denominator of our rational expression, we can cancel them out, thereby simplifying the expression.
Let's begin by examining the numerator: 3x^2 - 12. Notice that both terms share a common factor of 3. We can factor out the 3, resulting in:
3(x^2 - 4)
Now, observe the expression inside the parentheses: x^2 - 4. This is a classic example of a difference of squares, a special factoring pattern that takes the form a^2 - b^2. The difference of squares can always be factored as (a + b)(a - b). In our case, x^2 - 4 fits this pattern perfectly, with a = x and b = 2. Therefore, we can factor x^2 - 4 as (x + 2)(x - 2). Consequently, the fully factored numerator becomes:
3(x + 2)(x - 2)
Now, let's turn our attention to the denominator: 10 - 5x. Again, we can identify a common factor, this time 5. Factoring out the 5, we get:
5(2 - x)
Canceling Common Factors: The Simplification Step
With both the numerator and denominator factored, our rational expression now looks like this:
[3(x + 2)(x - 2)] / [5(2 - x)]
Here's where the magic of simplification happens. We look for common factors present in both the numerator and denominator. At first glance, it might seem like there are no identical factors. However, notice the terms (x - 2) in the numerator and (2 - x) in the denominator. These are very similar, differing only in the sign. We can manipulate one of these terms to match the other by factoring out a -1.
Let's factor out a -1 from the (2 - x) term in the denominator:
2 - x = -1(x - 2)
Now our expression becomes:
[3(x + 2)(x - 2)] / [5(-1)(x - 2)]
We now have a clear common factor of (x - 2) in both the numerator and denominator. We can cancel these out:
[3(x + 2)(x - 2)] / [5(-1)(x - 2)] = [3(x + 2)] / [5(-1)]
The Final Simplified Form
After canceling the common factor, we are left with:
[3(x + 2)] / -5
To present the simplified expression in its most conventional form, we can distribute the 3 in the numerator and move the negative sign to the front of the fraction:
(-3(x + 2)) / 5
Further distributing the -3, we arrive at the final simplified form:
(-3x - 6) / 5
Therefore, the simplified form of the rational expression (3x^2 - 12) / (10 - 5x) is (-3x - 6) / 5. This simplified form is equivalent to the original expression but is more concise and easier to work with in mathematical operations. Factoring and canceling common factors are essential techniques in simplifying rational expressions, enabling us to manipulate and understand complex mathematical relationships more effectively.
While we have successfully simplified the rational expression, it's crucial to remember the concept of domain restrictions. Domain restrictions arise in rational expressions because division by zero is undefined in mathematics. Therefore, any value of x that makes the denominator of the original expression equal to zero must be excluded from the domain.
In our original expression, (3x^2 - 12) / (10 - 5x), the denominator is (10 - 5x). To find the values of x that make the denominator zero, we set the denominator equal to zero and solve for x:
10 - 5x = 0
Subtract 10 from both sides:
-5x = -10
Divide both sides by -5:
x = 2
This tells us that x = 2 makes the original denominator zero. Therefore, x = 2 must be excluded from the domain of the expression. We can express this domain restriction in various ways:
- x ≠2
- All real numbers except 2
- (-∞, 2) ∪ (2, ∞) (interval notation)
It's essential to state the domain restriction alongside the simplified expression to provide a complete and accurate representation of the mathematical function. The simplified expression (-3x - 6) / 5 is equivalent to the original expression for all values of x except x = 2. At x = 2, the original expression is undefined, while the simplified expression is defined. Therefore, stating the domain restriction ensures we acknowledge this difference and maintain mathematical rigor.
Simplifying rational expressions is not merely a mathematical exercise; it's a fundamental skill with far-reaching implications in various areas of mathematics and beyond. Understanding and applying simplification techniques allows us to:
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Solve Equations More Easily: Simplified expressions are much easier to manipulate when solving equations. By reducing the complexity of the equation, we can isolate the variable and find its value more efficiently.
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Graph Functions Accurately: When graphing rational functions, the simplified form helps us identify key features such as intercepts, asymptotes, and holes. These features determine the shape and behavior of the graph, allowing for accurate visualization of the function.
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Perform Calculus Operations: In calculus, simplifying expressions is often a necessary step before differentiation or integration. Simplified forms make these operations more manageable and less prone to errors.
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Model Real-World Phenomena: Mathematical models often involve complex expressions. Simplifying these expressions allows us to analyze and interpret the model more effectively, gaining insights into the real-world phenomena being represented.
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Improve Mathematical Communication: Presenting mathematical results in simplified form demonstrates a clear understanding of the concepts and facilitates communication with others. A simplified expression is easier to understand and interpret, making it easier to share mathematical knowledge.
In conclusion, mastering the art of simplifying rational expressions is a crucial investment in your mathematical journey. It empowers you to tackle complex problems, gain deeper insights, and communicate mathematical ideas with clarity and precision. The techniques we've explored in this article, including factoring, canceling common factors, and identifying domain restrictions, form the bedrock of this essential skill. Keep practicing and applying these techniques, and you'll find your mathematical abilities growing exponentially.
To solidify your understanding of simplifying rational expressions, here are some avenues for further exploration and practice:
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Work Through Additional Examples: Seek out a variety of rational expression simplification problems. Start with simpler examples and gradually progress to more complex ones. This will help you build confidence and develop your problem-solving skills.
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Consult Textbooks and Online Resources: Many textbooks and websites offer detailed explanations and examples of simplifying rational expressions. Take advantage of these resources to deepen your understanding and explore different approaches.
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Practice Factoring Techniques: Factoring is the foundation of simplifying rational expressions. Practice factoring different types of polynomials, including quadratic expressions, difference of squares, and sum/difference of cubes.
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Pay Attention to Domain Restrictions: Always remember to identify and state the domain restrictions when simplifying rational expressions. This is a crucial step in ensuring mathematical accuracy.
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Seek Feedback from Others: Discuss your solutions with classmates, teachers, or online forums. Getting feedback from others can help you identify areas for improvement and gain new perspectives.
By actively engaging with these resources and practicing consistently, you'll develop a strong foundation in simplifying rational expressions and unlock new possibilities in your mathematical explorations.
