Which Of The Following Is Equal To The Rational Expression When $x \neq -4$ Or $3$?$\frac{x^2 - 4x + 3}{x^2 + X - 12}$A. $\frac{x+1}{x+4}$ B. $\frac{x-1}{x+4}$ C. $\frac{x-1}{x-3}$ D.

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Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying rational expressions, with a focus on the given problem: x2āˆ’4x+3x2+xāˆ’12\frac{x^2 - 4x + 3}{x^2 + x - 12}. We will examine the options provided and determine which one is equal to the rational expression when xā‰ āˆ’4x \neq -4 or 33.

Understanding Rational Expressions

A rational expression is a fraction that contains variables and/or constants in the numerator and denominator. It is essential to understand that a rational expression can be simplified by canceling out common factors between the numerator and denominator.

Step 1: Factor the Numerator and Denominator

To simplify the given rational expression, we need to factor the numerator and denominator.

x2āˆ’4x+3x2+xāˆ’12\frac{x^2 - 4x + 3}{x^2 + x - 12}

We can factor the numerator as follows:

x2āˆ’4x+3=(xāˆ’3)(xāˆ’1)x^2 - 4x + 3 = (x - 3)(x - 1)

And the denominator can be factored as:

x2+xāˆ’12=(x+4)(xāˆ’3)x^2 + x - 12 = (x + 4)(x - 3)

Step 2: Cancel Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out common factors.

(xāˆ’3)(xāˆ’1)(x+4)(xāˆ’3)\frac{(x - 3)(x - 1)}{(x + 4)(x - 3)}

We can cancel out the common factor (xāˆ’3)(x - 3):

xāˆ’1x+4\frac{x - 1}{x + 4}

Evaluating the Options

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

A. x+1x+4\frac{x+1}{x+4}

B. xāˆ’1x+4\frac{x-1}{x+4}

C. xāˆ’1xāˆ’3\frac{x-1}{x-3}

D. x+1xāˆ’3\frac{x+1}{x-3}

Based on our simplification, we can see that option B is equal to the rational expression when xā‰ āˆ’4x \neq -4 or 33.

Conclusion

In conclusion, simplifying rational expressions is a crucial skill in algebra. By following the steps outlined in this article, we can simplify the given rational expression and determine which option is equal to it when xā‰ āˆ’4x \neq -4 or 33. The correct answer is option B: xāˆ’1x+4\frac{x-1}{x+4}.

Final Answer

The final answer is B\boxed{B}.

Additional Tips and Resources

  • To simplify rational expressions, it is essential to factor the numerator and denominator.
  • Cancel out common factors between the numerator and denominator.
  • Be careful when canceling out common factors, as it is essential to ensure that the factor is not equal to zero.
  • Practice simplifying rational expressions with different problems to become proficient in this skill.

References

  • [Algebra Book] by [Author]
  • [Online Resource] by [Website]

Related Topics

  • Simplifying rational expressions
  • Factoring polynomials
  • Canceling out common factors

Frequently Asked Questions

  • Q: How do I simplify rational expressions? A: To simplify rational expressions, factor the numerator and denominator and cancel out common factors.
  • Q: What is the correct answer to the given problem? A: The correct answer is option B: xāˆ’1x+4\frac{x-1}{x+4}.
  • Q: What are some additional tips for simplifying rational expressions? A: Be careful when canceling out common factors, and practice simplifying rational expressions with different problems.
    Frequently Asked Questions: Simplifying Rational Expressions ===========================================================

Introduction

Simplifying rational expressions is a crucial skill in algebra, and it can be a bit challenging for some students. In this article, we will address some of the most frequently asked questions about simplifying rational expressions.

Q: What is a rational expression?

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

Q: How do I simplify a rational expression?

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

Q: What is factoring?

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

Q: How do I factor a polynomial?

There are several methods to factor a polynomial, including:

  • Factoring out the greatest common factor (GCF)
  • Factoring by grouping
  • Factoring quadratic expressions

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

The greatest common factor (GCF) is the largest factor that divides each term of a polynomial.

Q: How do I cancel out common factors?

To cancel out common factors, you need to identify the common factors between the numerator and denominator and divide both the numerator and denominator by that factor.

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

Some common mistakes to avoid when simplifying rational expressions include:

  • Not factoring the numerator and denominator
  • Not canceling out common factors
  • Canceling out a factor that is equal to zero

Q: How do I know if a factor is equal to zero?

To determine if a factor is equal to zero, you need to set the factor equal to zero and solve for the variable.

Q: What are some additional tips for simplifying rational expressions?

Some additional tips for simplifying rational expressions include:

  • Practice simplifying rational expressions with different problems
  • Use a calculator to check your work
  • Read the problem carefully and understand what is being asked

Q: How do I know if a rational expression is in its simplest form?

A rational expression is in its simplest form when there are no common factors between the numerator and denominator.

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

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

  • Calculating probabilities
  • Modeling population growth
  • Analyzing data

Conclusion

In conclusion, simplifying rational expressions is a crucial skill in algebra, and it requires practice and patience. By following the steps outlined in this article, you can simplify rational expressions and avoid common mistakes.

Final Answer

The final answer is B\boxed{B}.

Additional Resources

  • [Algebra Book] by [Author]
  • [Online Resource] by [Website]

Related Topics

  • Simplifying rational expressions
  • Factoring polynomials
  • Canceling out common factors

Frequently Asked Questions

  • Q: How do I simplify a rational expression? A: To simplify a rational expression, you need to factor the numerator and denominator and cancel out common factors.
  • Q: What is factoring? A: Factoring is the process of expressing a polynomial as a product of simpler polynomials.
  • Q: How do I cancel out common factors? A: To cancel out common factors, you need to identify the common factors between the numerator and denominator and divide both the numerator and denominator by that factor.