Which Of The Following Is Equal To The Rational Expression When $x \neq 1$ Or 3? X 2 − 9 ( X − 1 ) ( X − 3 ) \frac{x^2-9}{(x-1)(x-3)} ( X − 1 ) ( X − 3 ) X 2 − 9 ​ A. X − 3 X − 1 \frac{x-3}{x-1} X − 1 X − 3 ​ B. X + 3 X − 1 \frac{x+3}{x-1} X − 1 X + 3 ​ C. X − 1 X − 3 \frac{x-1}{x-3} X − 3 X − 1 ​

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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 the process of simplifying rational expressions, with a focus on the given problem: x2−9(x−1)(x−3)\frac{x^2-9}{(x-1)(x-3)}. We will examine the options provided and determine which one is equal to the given rational expression when x≠1x \neq 1 or 3.

Understanding 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 factoring the numerator and denominator, canceling out common factors, and then simplifying the resulting expression.

The Given Rational Expression

The given rational expression is x2−9(x−1)(x−3)\frac{x^2-9}{(x-1)(x-3)}. To simplify this expression, we need to factor the numerator and denominator.

Factoring the Numerator

The numerator, x2−9x^2-9, can be factored as a difference of squares:

x2−9=(x−3)(x+3)x^2-9 = (x-3)(x+3)

Factoring the Denominator

The denominator, (x−1)(x−3)(x-1)(x-3), is already factored.

Simplifying the Rational Expression

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

(x−3)(x+3)(x−1)(x−3)=x+3x−1\frac{(x-3)(x+3)}{(x-1)(x-3)} = \frac{x+3}{x-1}

Evaluating the Options

Now that we have simplified the rational expression, we can evaluate the options provided:

A. x−3x−1\frac{x-3}{x-1} B. x+3x−1\frac{x+3}{x-1} C. x−1x−3\frac{x-1}{x-3}

Based on our simplification, we can see that option B, x+3x−1\frac{x+3}{x-1}, is equal to the given rational expression when x≠1x \neq 1 or 3.

Conclusion

In conclusion, simplifying rational expressions is a crucial skill for any math enthusiast. By factoring the numerator and denominator, canceling out common factors, and then simplifying the resulting expression, we can simplify rational expressions and determine which option is equal to the given rational expression. In this case, option B, x+3x−1\frac{x+3}{x-1}, is equal to the given rational expression when x≠1x \neq 1 or 3.

Additional Tips and Tricks

  • When simplifying rational expressions, always factor the numerator and denominator.
  • Cancel out common factors to simplify the expression.
  • Be careful when canceling out common factors, as this can result in an incorrect simplification.
  • Always check your work by plugging in values for the variable to ensure that the simplified expression is equal to the original expression.

Common Mistakes to Avoid

  • Failing to factor the numerator and denominator.
  • Canceling out common factors incorrectly.
  • Not checking your work by plugging in values for the variable.

Real-World Applications

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

  • Calculating probabilities and statistics.
  • Modeling population growth and decline.
  • Analyzing financial data and making informed investment decisions.

Final Thoughts

Introduction

In our previous article, we explored the process of simplifying rational expressions, with a focus on the given problem: x2−9(x−1)(x−3)\frac{x^2-9}{(x-1)(x-3)}. We also discussed the importance of simplifying rational expressions in various real-world applications. In this article, we will provide a Q&A guide to help you better understand the concept of 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: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, cancel out common factors, and then simplify the resulting expression.

Q: What is factoring?

A: Factoring is the process of expressing a polynomial as a product of simpler polynomials.

Q: How do I factor a polynomial?

A: There are several methods to factor a polynomial, including:

  • Factoring out the greatest common factor (GCF)
  • Factoring by grouping
  • Factoring by difference of squares
  • Factoring by difference of cubes

Q: What is the difference of squares?

A: The difference of squares is a polynomial that can be factored as:

a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)

Q: What is the difference of cubes?

A: The difference of cubes is a polynomial that can be factored as:

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Q: How do I cancel out common factors?

A: To cancel out common factors, you need to identify the common factors in the numerator and denominator and cancel them out.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides both the numerator and denominator.

Q: How do I find the GCF?

A: To find the GCF, you need to list the factors of the numerator and denominator and identify the largest factor that divides both.

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
  • Canceling out common factors incorrectly
  • Not checking your work by plugging in values for the variable

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

A: To check your work, you need to plug in values for the variable and verify that the simplified expression is equal to the original expression.

Q: What are some real-world applications of simplifying rational expressions?

A: Simplifying rational expressions has numerous real-world applications, including:

  • Calculating probabilities and statistics
  • Modeling population growth and decline
  • Analyzing financial data and making informed investment decisions

Conclusion

In conclusion, simplifying rational expressions is a crucial skill for any math enthusiast. By following the steps outlined in this article, you can simplify rational expressions and determine which option is equal to the given rational expression. Remember to always factor the numerator and denominator, cancel out common factors, and check your work by plugging in values for the variable. With practice and patience, you can become proficient in simplifying rational expressions and apply this skill to real-world problems.

Additional Resources

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

Final Thoughts

Simplifying rational expressions is a crucial skill for any math enthusiast. By following the steps outlined in this article, you can simplify rational expressions and determine which option is equal to the given rational expression. Remember to always factor the numerator and denominator, cancel out common factors, and check your work by plugging in values for the variable. With practice and patience, you can become proficient in simplifying rational expressions and apply this skill to real-world problems.