Simplify The Expression:$\[ \frac{\frac{x^2+4x+3}{2x-1}}{\frac{x^2+x}{2x^2-3x+1}} \\]Choose The Correct Simplification:A. \[$\frac{(x+3)(x-1)}{x}\$\]B. \[$\frac{(x+3)(x+1)}{x(2x-1)}\$\]C.

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Introduction

Simplifying complex expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves breaking down intricate expressions into simpler forms, making them easier to understand and work with. In this article, we will focus on simplifying a specific expression involving fractions and polynomials.

The Given Expression

The expression we need to simplify is:

x2+4x+32x−1x2+x2x2−3x+1\frac{\frac{x^2+4x+3}{2x-1}}{\frac{x^2+x}{2x^2-3x+1}}

Step 1: Factor the Numerators and Denominators

To simplify the expression, we first need to factor the numerators and denominators of the fractions.

  • The numerator of the first fraction, x2+4x+3x^2+4x+3, can be factored as (x+3)(x+1)(x+3)(x+1).
  • The denominator of the first fraction, 2x−12x-1, remains the same.
  • The numerator of the second fraction, x2+xx^2+x, can be factored as x(x+1)x(x+1).
  • The denominator of the second fraction, 2x2−3x+12x^2-3x+1, can be factored as (2x−1)(x−1)(2x-1)(x-1).

Step 2: Rewrite the Expression with Factored Numerators and Denominators

Now that we have factored the numerators and denominators, we can rewrite the expression as:

(x+3)(x+1)2x−1x(x+1)(2x−1)(x−1)\frac{\frac{(x+3)(x+1)}{2x-1}}{\frac{x(x+1)}{(2x-1)(x-1)}}

Step 3: Cancel Common Factors

We can simplify the expression further by canceling common factors in the numerator and denominator.

  • The factor (x+1)(x+1) appears in both the numerator and denominator, so we can cancel it out.
  • The factor (2x−1)(2x-1) appears in both the numerator and denominator, so we can cancel it out.

After canceling the common factors, the expression becomes:

x+3x(x−1)\frac{x+3}{x(x-1)}

Step 4: Simplify the Expression

We can simplify the expression further by multiplying the numerator and denominator by the reciprocal of the denominator.

  • The reciprocal of the denominator is 1x(x−1)\frac{1}{x(x-1)}.
  • Multiplying the numerator and denominator by the reciprocal of the denominator, we get:

(x+3)(x−1)x\frac{(x+3)(x-1)}{x}

Conclusion

In conclusion, the simplified expression is:

(x+3)(x−1)x\frac{(x+3)(x-1)}{x}

This is the correct simplification of the given expression.

Discussion

The correct simplification of the expression is:

(x+3)(x−1)x\frac{(x+3)(x-1)}{x}

This simplification involves factoring the numerators and denominators, canceling common factors, and multiplying the numerator and denominator by the reciprocal of the denominator.

Comparison with Other Options

Let's compare the correct simplification with the other options:

  • Option A: (x+3)(x+1)x\frac{(x+3)(x+1)}{x}
  • Option B: (x+3)(x+1)x(2x−1)\frac{(x+3)(x+1)}{x(2x-1)}
  • Option C: (not provided)

The correct simplification is different from Option A, which has an extra factor of (x+1)(x+1) in the numerator. The correct simplification is also different from Option B, which has an extra factor of (2x−1)(2x-1) in the denominator.

Final Answer

The final answer is:

(x+3)(x−1)x\frac{(x+3)(x-1)}{x}

This is the correct simplification of the given expression.

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Introduction

In our previous article, we explored the process of simplifying complex expressions involving fractions and polynomials. We walked through a step-by-step guide to simplify the expression:

x2+4x+32x−1x2+x2x2−3x+1\frac{\frac{x^2+4x+3}{2x-1}}{\frac{x^2+x}{2x^2-3x+1}}

In this article, we will address some common questions and concerns related to simplifying complex expressions.

Q&A

Q: What is the first step in simplifying a complex expression?

A: The first step in simplifying a complex expression is to factor the numerators and denominators of the fractions involved.

Q: How do I factor the numerators and denominators?

A: To factor the numerators and denominators, look for common factors or use algebraic techniques such as factoring quadratics or grouping.

Q: What if I have a fraction with a variable in the denominator?

A: If you have a fraction with a variable in the denominator, you can simplify it by canceling out any common factors between the numerator and denominator.

Q: Can I simplify an expression with multiple fractions?

A: Yes, you can simplify an expression with multiple fractions by following the same steps as before: factor the numerators and denominators, cancel out common factors, and multiply the numerator and denominator by the reciprocal of the denominator.

Q: What if I get stuck or unsure about how to simplify an expression?

A: If you get stuck or unsure about how to simplify an expression, try breaking it down into smaller parts, using algebraic techniques, or seeking help from a teacher or tutor.

Q: Are there any common mistakes to avoid when simplifying complex expressions?

A: Yes, some common mistakes to avoid when simplifying complex expressions include:

  • Not factoring the numerators and denominators
  • Not canceling out common factors
  • Not multiplying the numerator and denominator by the reciprocal of the denominator
  • Not checking for any remaining common factors

Tips and Tricks

Tip 1: Start with the simplest expression

When simplifying complex expressions, start with the simplest expression and work your way up. This will help you avoid getting overwhelmed and make the process more manageable.

Tip 2: Use algebraic techniques

Use algebraic techniques such as factoring quadratics or grouping to simplify complex expressions.

Tip 3: Check for common factors

Always check for common factors between the numerator and denominator before simplifying an expression.

Tip 4: Multiply the numerator and denominator by the reciprocal of the denominator

When simplifying an expression, multiply the numerator and denominator by the reciprocal of the denominator to eliminate any remaining common factors.

Conclusion

Simplifying complex expressions can be a challenging task, but with practice and patience, you can master the skills and techniques needed to simplify even the most intricate expressions. Remember to factor the numerators and denominators, cancel out common factors, and multiply the numerator and denominator by the reciprocal of the denominator. With these tips and tricks, you'll be well on your way to becoming a pro at simplifying complex expressions.

Final Answer

The final answer is:

(x+3)(x−1)x\frac{(x+3)(x-1)}{x}

This is the correct simplification of the given expression.