Simplify The Expression:$\[ \frac{x^2 - 4x - 21}{x^2 - 49} \\]

Feb 28, 2025 by ADMIN 63 views

Simplify the Expression: A Step-by-Step Guide to Factoring and Reducing Rational Expressions

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems more efficiently and accurately. One of the most common types of expressions that require simplification is rational expressions, which are fractions that contain variables and constants in the numerator and denominator. In this article, we will focus on simplifying the expression $\frac{x^2 - 4x - 21}{x^2 - 49}$ using factoring and reducing techniques.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at its components. The numerator is a quadratic expression $x^2 - 4x - 21$ , while the denominator is also a quadratic expression $x^2 - 49$ . Our goal is to simplify this expression by factoring and reducing it to its simplest form.

Factoring the Numerator

To simplify the expression, we need to start by factoring the numerator. The numerator is a quadratic expression that can be factored using the method of splitting the middle term. This method involves finding two numbers whose product is equal to the product of the coefficient of the squared term and the constant term, and whose sum is equal to the coefficient of the linear term.

Let's apply this method to the numerator:

$x^2 - 4x - 21$

We need to find two numbers whose product is equal to $-21$ and whose sum is equal to $-4$ . After some trial and error, we find that the numbers are $-7$ and $3$ , since $-7 \times 3 = -21$ and $-7 + 3 = -4$ .

Now, we can rewrite the numerator as:

$x^2 - 7x + 3x - 21$

We can then factor by grouping:

$(x^2 - 7x) + (3x - 21)$

Factoring out the common terms, we get:

$x(x - 7) + 3(x - 7)$

Now, we can factor out the common binomial factor $(x - 7)$ :

$(x + 3)(x - 7)$

So, the factored form of the numerator is $(x + 3)(x - 7)$ .

Factoring the Denominator

Next, we need to factor the denominator. The denominator is a quadratic expression $x^2 - 49$ , which can be factored as a difference of squares:

$x^2 - 49 = (x + 7)(x - 7)$

Simplifying the Expression

Now that we have factored the numerator and denominator, we can simplify the expression by canceling out any common factors.

The expression is:

$\frac{(x + 3)(x - 7)}{(x + 7)(x - 7)}$

We can see that the numerator and denominator have a common factor of $(x - 7)$ . We can cancel out this common factor:

$\frac{(x + 3)}{(x + 7)}$

So, the simplified form of the expression is $\frac{x + 3}{x + 7}$ .

Conclusion

In this article, we simplified the expression $\frac{x^2 - 4x - 21}{x^2 - 49}$ using factoring and reducing techniques. We factored the numerator and denominator, and then canceled out any common factors to simplify the expression. The simplified form of the expression is $\frac{x + 3}{x + 7}$ .

Final Answer

The final answer is $\boxed{\frac{x + 3}{x + 7}}$ .

Additional Tips and Tricks

When simplifying rational expressions, always look for common factors in the numerator and denominator.
Use factoring techniques such as splitting the middle term and difference of squares to simplify the numerator and denominator.
Cancel out any common factors to simplify the expression.
Always check your work by plugging in values for the variable to ensure that the simplified expression is equivalent to the original expression.

Common Mistakes to Avoid

Failing to factor the numerator and denominator properly.
Not canceling out common factors.
Not checking the work by plugging in values for the variable.

Real-World Applications

Simplifying rational expressions has many real-world applications in fields such as engineering, economics, and computer science. For example, in engineering, simplifying rational expressions can help us design and optimize systems, while in economics, it can help us model and analyze economic systems. In computer science, simplifying rational expressions can help us develop more efficient algorithms and data structures.

Final Thoughts

Simplifying rational expressions is an essential skill in mathematics that has many real-world applications. By following the steps outlined in this article, you can simplify rational expressions and develop a deeper understanding of algebraic expressions. Remember to always look for common factors, use factoring techniques, and cancel out common factors to simplify the expression. With practice and patience, you can become proficient in simplifying rational expressions and tackle more complex problems with confidence.

Introduction

In our previous article, we explored the process of simplifying rational expressions using factoring and reducing techniques. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in simplifying rational expressions.

Q&A: Simplifying Rational Expressions

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables and constants in the numerator and denominator.

Q: Why is it important to simplify rational expressions?

A: Simplifying rational expressions is important because it helps us to:

Reduce the complexity of the expression
Make it easier to work with
Avoid errors and mistakes
Solve problems more efficiently and accurately

Q: What are the steps involved in simplifying a rational expression?

A: The steps involved in simplifying a rational expression are:

Factor the numerator and denominator
Cancel out any common factors
Simplify the expression

Q: How do I factor the numerator and denominator?

A: To factor the numerator and denominator, you can use various techniques such as:

Splitting the middle term
Difference of squares
Greatest common factor (GCF)

Q: What is the difference between factoring and simplifying a rational expression?

A: Factoring a rational expression involves breaking it down into its simplest form, while simplifying a rational expression involves reducing it to its simplest form by canceling out common factors.

Q: Can I simplify a rational expression that has no common factors?

A: Yes, you can still simplify a rational expression that has no common factors by reducing it to its simplest form.

Q: How do I know if a rational expression is already simplified?

A: A rational expression is already simplified if it has no common factors between the numerator and denominator.

Q: Can I simplify a rational expression with a variable in the denominator?

A: Yes, you can simplify a rational expression with a variable in the denominator by factoring and canceling out common factors.

Q: What are some common mistakes to avoid when simplifying rational expressions?

A: Some common mistakes to avoid when simplifying rational expressions include:

Failing to factor the numerator and denominator properly
Not canceling out common factors
Not checking the work by plugging in values for the variable

Q: How do I check my work when simplifying a rational expression?

A: To check your work, plug in values for the variable and simplify the expression. If the simplified expression is equivalent to the original expression, then your work is correct.

Real-World Applications

Final Thoughts

Simplifying rational expressions is an essential skill in mathematics that has many real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can simplify rational expressions and develop a deeper understanding of algebraic expressions. Remember to always look for common factors, use factoring techniques, and cancel out common factors to simplify the expression. With practice and patience, you can become proficient in simplifying rational expressions and tackle more complex problems with confidence.

Additional Resources

Khan Academy: Simplifying Rational Expressions
Mathway: Simplifying Rational Expressions
Wolfram Alpha: Simplifying Rational Expressions

Final Answer

The final answer is $\boxed{\frac{x + 3}{x + 7}}$ .