Simplify The Rational Expression By Canceling Common Factors:$\[ \frac{x^2+6x+8}{x^2+3x-4} \\]Options:A. \[$\frac{x-1}{x+2}\$\]B. \[$\frac{x+2}{x-1}\$\]C. \[$\frac{x+4}{x-1}\$\]D. \[$\frac{x+2}{x+4}\$\]

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Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore how to simplify a rational expression by canceling common factors. We will use a step-by-step approach to break down the problem and provide a clear understanding of the process.

What are Rational Expressions?

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by canceling common factors, which is a process of dividing both the numerator and the denominator by a common factor.

Step 1: Factor the Numerator and Denominator

To simplify a rational expression, we need to factor the numerator and the denominator. Factoring involves breaking down an expression into its simplest form by identifying common factors.

Let's take the given rational expression:

{ \frac{x^2+6x+8}{x^2+3x-4} \}

We need to factor the numerator and the denominator:

Numerator: x2+6x+8x^2+6x+8

Denominator: x2+3x−4x^2+3x-4

Factoring the Numerator

The numerator can be factored as:

x2+6x+8=(x+4)(x+2)x^2+6x+8 = (x+4)(x+2)

Factoring the Denominator

The denominator can be factored as:

x2+3x−4=(x+4)(x−1)x^2+3x-4 = (x+4)(x-1)

Step 2: Cancel Common Factors

Now that we have factored the numerator and the denominator, we can cancel common factors. In this case, we have a common factor of (x+4)(x+4) in both the numerator and the denominator.

We can cancel this common factor by dividing both the numerator and the denominator by (x+4)(x+4):

{ \frac{(x+4)(x+2)}{(x+4)(x-1)} = \frac{x+2}{x-1} \}

Step 3: Simplify the Expression

Now that we have canceled the common factor, we can simplify the expression by canceling any remaining common factors.

In this case, we have a common factor of (x+2)(x+2) in both the numerator and the denominator, but we have already canceled this factor in the previous step.

Therefore, the simplified expression is:

{ \frac{x+2}{x-1} \}

Conclusion

Simplifying a rational expression by canceling common factors is a straightforward process that involves factoring the numerator and the denominator and canceling common factors. By following the steps outlined in this article, you can simplify any rational expression and arrive at the correct solution.

Answer

The correct answer is:

{ \frac{x+2}{x-1} \}

This is option B.

Discussion

The discussion category for this problem is mathematics.

Additional Tips and Tricks

Here are some additional tips and tricks to help you simplify rational expressions:

  • Always factor the numerator and the denominator before canceling common factors.
  • Make sure to cancel common factors in both the numerator and the denominator.
  • Simplify the expression by canceling any remaining common factors.
  • Check your work by plugging in values for the variables to ensure that the expression is true.

By following these tips and tricks, you can simplify rational expressions with ease and arrive at the correct solution.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying rational expressions:

  • Failing to factor the numerator and the denominator.
  • Failing to cancel common factors in both the numerator and the denominator.
  • Not simplifying the expression by canceling any remaining common factors.
  • Not checking your work by plugging in values for the variables.

By avoiding these common mistakes, you can simplify rational expressions with confidence and arrive at the correct solution.

Real-World Applications

Simplifying rational expressions has many real-world applications, including:

  • Algebra: Simplifying rational expressions is a crucial skill for any math enthusiast, and it is used extensively in algebra.
  • Calculus: Simplifying rational expressions is used extensively in calculus, particularly in the study of limits and derivatives.
  • Physics: Simplifying rational expressions is used extensively in physics, particularly in the study of motion and energy.
  • Engineering: Simplifying rational expressions is used extensively in engineering, particularly in the study of circuits and systems.

By understanding how to simplify rational expressions, you can apply this knowledge to a wide range of real-world problems and arrive at the correct solution.

Conclusion

Introduction

In our previous article, we explored how to simplify a rational expression by canceling common factors. We covered the steps involved in factoring the numerator and the denominator, canceling common factors, and simplifying the expression. In this article, we will answer some frequently asked questions (FAQs) related to simplifying rational expressions.

Q: What is a rational expression?

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

Q: Why do we need to simplify rational expressions?

A: We need to simplify rational expressions to make them easier to work with and to arrive at the correct solution. Simplifying rational expressions involves canceling common factors, which can make the expression more manageable and easier to understand.

Q: How do I factor the numerator and the denominator?

A: To factor the numerator and the denominator, you need to identify the common factors and break down the expression into its simplest form. You can use various factoring techniques, such as factoring by grouping, factoring by difference of squares, and factoring by greatest common factor.

Q: What is the difference between factoring and canceling common factors?

A: Factoring involves breaking down an expression into its simplest form by identifying common factors, while canceling common factors involves dividing both the numerator and the denominator by a common factor.

Q: Can I cancel common factors in both the numerator and the denominator?

A: Yes, you can cancel common factors in both the numerator and the denominator. However, make sure to cancel common factors in both the numerator and the denominator to avoid any errors.

Q: How do I know if I have canceled all the common factors?

A: To ensure that you have canceled all the common factors, you need to check your work by plugging in values for the variables. If the expression is true for all values of the variables, then you have canceled all the 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 the denominator.
  • Failing to cancel common factors in both the numerator and the denominator.
  • Not simplifying the expression by canceling any remaining common factors.
  • Not checking your work by plugging in values for the variables.

Q: How do I apply simplifying rational expressions to real-world problems?

A: Simplifying rational expressions has many real-world applications, including algebra, calculus, physics, and engineering. By understanding how to simplify rational expressions, you can apply this knowledge to a wide range of real-world problems and arrive at the correct solution.

Q: Can I use technology to simplify rational expressions?

A: Yes, you can use technology to simplify rational expressions. Many graphing calculators and computer algebra systems (CAS) can simplify rational expressions and provide the correct solution.

Conclusion

Simplifying rational expressions by canceling common factors is a crucial skill for any math enthusiast. By following the steps outlined in this article and answering the FAQs, you can simplify any rational expression and arrive at the correct solution. Remember to always factor the numerator and the denominator, cancel common factors, and simplify the expression by canceling any remaining common factors. With practice and patience, you can become proficient in simplifying rational expressions and apply this knowledge to a wide range of real-world problems.

Additional Resources

Here are some additional resources to help you simplify rational expressions:

  • Khan Academy: Simplifying Rational Expressions
  • Mathway: Simplifying Rational Expressions
  • Wolfram Alpha: Simplifying Rational Expressions
  • MIT OpenCourseWare: Simplifying Rational Expressions

By using these resources and practicing simplifying rational expressions, you can become proficient in this skill and apply it to a wide range of real-world problems.