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

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Introduction

Simplifying complex algebraic 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 work with and understand. 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 both 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 between the numerator and denominator.

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

  • The denominator, x(x−1)x(x-1), can be factored as x(x−1)x(x-1).

However, we can simplify the expression further by multiplying the numerator and denominator by the reciprocal of the denominator.

x+3x(x−1)⋅11\frac{x+3}{x(x-1)} \cdot \frac{1}{1}

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

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

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

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\frac{(x+3<br/> # Simplifying Complex Algebraic Expressions: A Q&A Guide =========================================================== ## Introduction ---------------- Simplifying complex algebraic 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 work with and understand. In this article, we will focus on simplifying a specific expression involving fractions and polynomials. ## Q&A: Simplifying Complex Algebraic Expressions ---------------------------------------------- ### Q: What is the first step in simplifying a complex algebraic expression? A: The first step in simplifying a complex algebraic expression is to factor the numerators and denominators of both fractions. ### Q: How do I factor the numerators and denominators? A: To factor the numerators and denominators, look for common factors and use the distributive property to break down the expressions. ### Q: What is the distributive property? A: The distributive property is a mathematical property that allows us to multiply a single term by multiple terms. It states that for any numbers a, b, and c: a(b + c) = ab + ac ### Q: How do I cancel common factors? A: To cancel common factors, look for terms that appear in both the numerator and denominator. If a term appears in both the numerator and denominator, you can cancel it out. ### Q: What is the final simplified expression? A: The final simplified expression is: $\frac{(x+3)}{x(x-1)}

Q: Can I simplify the expression further?

A: Yes, you can simplify the expression further by multiplying the numerator and denominator by the reciprocal of the denominator.

Q: What is the reciprocal of the denominator?

A: The reciprocal of the denominator is 1 divided by the denominator. In this case, the reciprocal of the denominator is 1 divided by x(x-1).

Q: How do I multiply the numerator and denominator by the reciprocal of the denominator?

A: To multiply the numerator and denominator by the reciprocal of the denominator, multiply the numerator by the reciprocal of the denominator and multiply the denominator by the reciprocal of the denominator.

Q: What is the final simplified expression after multiplying the numerator and denominator by the reciprocal of the denominator?

A: The final simplified expression after multiplying the numerator and denominator by the reciprocal of the denominator is:

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

Q: Can I simplify the expression further?

A: Yes, you can simplify the expression further by canceling common factors.

Q: What is the final simplified expression after canceling common factors?

A: The final simplified expression after canceling common factors is:

(x+3)x(x−1)\frac{(x+3)}{x(x-1)}

Conclusion

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. By following the steps outlined in this article, you can simplify complex expressions and make them easier to work with and understand.

Common Mistakes to Avoid

  • Not factoring the numerators and denominators
  • Not canceling common factors
  • Not multiplying the numerator and denominator by the reciprocal of the denominator

Tips and Tricks

  • Use the distributive property to break down complex expressions
  • Look for common factors to cancel out
  • Multiply the numerator and denominator by the reciprocal of the denominator to simplify the expression further

Practice Problems

  • 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}}
  • Simplify the expression: x2−4x+3x−1x2−2x+1x2−3x+2\frac{\frac{x^2-4x+3}{x-1}}{\frac{x^2-2x+1}{x^2-3x+2}}
  • Simplify the expression: x2+2x+1x+1x2+x+1x2+2x+1\frac{\frac{x^2+2x+1}{x+1}}{\frac{x^2+x+1}{x^2+2x+1}}

Resources

  • Khan Academy: Algebra
  • Mathway: Algebra
  • Wolfram Alpha: Algebra

Conclusion

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. By following the steps outlined in this article, you can simplify complex expressions and make them easier to work with and understand. Remember to factor the numerators and denominators, cancel common factors, and multiply the numerator and denominator by the reciprocal of the denominator to simplify the expression further.