Simplify The Expression:${ \frac{2x^2 + 5x + 3}{x^2 - 3x - 4} \div \frac{4x^2 + 2x - 6}{x^2 - 8x + 16} }$A. { \frac{x-4}{2x-2}$}$B. { \frac{x-2}{x}$}$C. { \frac{2x-2}{x-4}$}$

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Introduction

In this article, we will simplify the given expression using algebraic techniques. The expression involves the division of two rational expressions, and we will use various methods to simplify it. We will start by understanding the properties of rational expressions and then apply the techniques to simplify the given expression.

Understanding Rational Expressions

A rational expression is a fraction in which the numerator and denominator are polynomials. Rational expressions can be simplified by factoring the numerator and denominator, canceling out common factors, and then simplifying the resulting expression.

Properties of Rational Expressions

  • A rational expression is equal to zero if and only if its numerator is equal to zero.
  • A rational expression is undefined if and only if its denominator is equal to zero.
  • Rational expressions can be added, subtracted, multiplied, and divided using the usual rules of arithmetic.

Simplifying the Given Expression

The given expression is:

2x2+5x+3x2−3x−4÷4x2+2x−6x2−8x+16\frac{2x^2 + 5x + 3}{x^2 - 3x - 4} \div \frac{4x^2 + 2x - 6}{x^2 - 8x + 16}

To simplify this expression, we will first factor the numerator and denominator of each rational expression.

Factoring the Numerator and Denominator

  • The numerator of the first rational expression can be factored as:

    2x2+5x+3=(2x+3)(x+1)2x^2 + 5x + 3 = (2x + 3)(x + 1)

  • The denominator of the first rational expression can be factored as:

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

  • The numerator of the second rational expression can be factored as:

    4x2+2x−6=(2x−3)(2x+2)4x^2 + 2x - 6 = (2x - 3)(2x + 2)

  • The denominator of the second rational expression can be factored as:

    x2−8x+16=(x−4)(x−4)x^2 - 8x + 16 = (x - 4)(x - 4)

Canceling Out Common Factors

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

  • The first rational expression can be simplified as:

    (2x+3)(x+1)(x−4)(x+1)=2x+3x−4\frac{(2x + 3)(x + 1)}{(x - 4)(x + 1)} = \frac{2x + 3}{x - 4}

  • The second rational expression can be simplified as:

    (2x−3)(2x+2)(x−4)(x−4)=2x−3x−4\frac{(2x - 3)(2x + 2)}{(x - 4)(x - 4)} = \frac{2x - 3}{x - 4}

Dividing the Rational Expressions

Now that we have simplified the rational expressions, we can divide them.

  • The given expression can be simplified as:

    2x+3x−4÷2x−3x−4=2x+3x−4×x−42x−3\frac{2x + 3}{x - 4} \div \frac{2x - 3}{x - 4} = \frac{2x + 3}{x - 4} \times \frac{x - 4}{2x - 3}

  • The resulting expression can be simplified as:

    (2x+3)(x−4)(x−4)(2x−3)=2x+32x−3\frac{(2x + 3)(x - 4)}{(x - 4)(2x - 3)} = \frac{2x + 3}{2x - 3}

Simplifying the Resulting Expression

The resulting expression can be simplified further by factoring the numerator and denominator.

  • The numerator of the resulting expression can be factored as:

    2x+3=(2x+3)2x + 3 = (2x + 3)

  • The denominator of the resulting expression can be factored as:

    2x−3=(2x−3)2x - 3 = (2x - 3)

  • The resulting expression can be simplified as:

    2x+32x−3\frac{2x + 3}{2x - 3}

Conclusion

In this article, we simplified the given expression using algebraic techniques. We started by understanding the properties of rational expressions and then applied the techniques to simplify the given expression. We factored the numerator and denominator of each rational expression, canceled out common factors, and then divided the rational expressions. The resulting expression was simplified further by factoring the numerator and denominator. The final answer is 2x+32x−3\boxed{\frac{2x + 3}{2x - 3}}.

Final Answer

The final answer is 2x+32x−3\boxed{\frac{2x + 3}{2x - 3}}.

=====================================================

Introduction

In this article, we will simplify the given expression using algebraic techniques. The expression involves the division of two rational expressions, and we will use various methods to simplify it. We will start by understanding the properties of rational expressions and then apply the techniques to simplify the given expression.

Understanding Rational Expressions

A rational expression is a fraction in which the numerator and denominator are polynomials. Rational expressions can be simplified by factoring the numerator and denominator, canceling out common factors, and then simplifying the resulting expression.

Properties of Rational Expressions

  • A rational expression is equal to zero if and only if its numerator is equal to zero.
  • A rational expression is undefined if and only if its denominator is equal to zero.
  • Rational expressions can be added, subtracted, multiplied, and divided using the usual rules of arithmetic.

Simplifying the Given Expression

The given expression is:

2x2+5x+3x2−3x−4÷4x2+2x−6x2−8x+16\frac{2x^2 + 5x + 3}{x^2 - 3x - 4} \div \frac{4x^2 + 2x - 6}{x^2 - 8x + 16}

To simplify this expression, we will first factor the numerator and denominator of each rational expression.

Factoring the Numerator and Denominator

  • The numerator of the first rational expression can be factored as:

    2x2+5x+3=(2x+3)(x+1)2x^2 + 5x + 3 = (2x + 3)(x + 1)

  • The denominator of the first rational expression can be factored as:

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

  • The numerator of the second rational expression can be factored as:

    4x2+2x−6=(2x−3)(2x+2)4x^2 + 2x - 6 = (2x - 3)(2x + 2)

  • The denominator of the second rational expression can be factored as:

    x2−8x+16=(x−4)(x−4)x^2 - 8x + 16 = (x - 4)(x - 4)

Canceling Out Common Factors

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

  • The first rational expression can be simplified as:

    (2x+3)(x+1)(x−4)(x+1)=2x+3x−4\frac{(2x + 3)(x + 1)}{(x - 4)(x + 1)} = \frac{2x + 3}{x - 4}

  • The second rational expression can be simplified as:

    (2x−3)(2x+2)(x−4)(x−4)=2x−3x−4\frac{(2x - 3)(2x + 2)}{(x - 4)(x - 4)} = \frac{2x - 3}{x - 4}

Dividing the Rational Expressions

Now that we have simplified the rational expressions, we can divide them.

  • The given expression can be simplified as:

    2x+3x−4÷2x−3x−4=2x+3x−4×x−42x−3\frac{2x + 3}{x - 4} \div \frac{2x - 3}{x - 4} = \frac{2x + 3}{x - 4} \times \frac{x - 4}{2x - 3}

  • The resulting expression can be simplified as:

    (2x+3)(x−4)(x−4)(2x−3)=2x+32x−3\frac{(2x + 3)(x - 4)}{(x - 4)(2x - 3)} = \frac{2x + 3}{2x - 3}

Simplifying the Resulting Expression

The resulting expression can be simplified further by factoring the numerator and denominator.

  • The numerator of the resulting expression can be factored as:

    2x+3=(2x+3)2x + 3 = (2x + 3)

  • The denominator of the resulting expression can be factored as:

    2x−3=(2x−3)2x - 3 = (2x - 3)

  • The resulting expression can be simplified as:

    2x+32x−3\frac{2x + 3}{2x - 3}

Q&A

Q: What is a rational expression?

A: A rational expression is a fraction in which the numerator and denominator are polynomials.

Q: How do you simplify a rational expression?

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

Q: What are the properties of rational expressions?

A: Rational expressions have the following properties:

  • A rational expression is equal to zero if and only if its numerator is equal to zero.
  • A rational expression is undefined if and only if its denominator is equal to zero.
  • Rational expressions can be added, subtracted, multiplied, and divided using the usual rules of arithmetic.

Q: How do you divide rational expressions?

A: To divide rational expressions, you can invert the second rational expression and multiply it by the first rational expression.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is 2x+32x−3\boxed{\frac{2x + 3}{2x - 3}}.

Conclusion

In this article, we simplified the given expression using algebraic techniques. We started by understanding the properties of rational expressions and then applied the techniques to simplify the given expression. We factored the numerator and denominator of each rational expression, canceled out common factors, and then divided the rational expressions. The resulting expression was simplified further by factoring the numerator and denominator. The final answer is 2x+32x−3\boxed{\frac{2x + 3}{2x - 3}}.

Final Answer

The final answer is 2x+32x−3\boxed{\frac{2x + 3}{2x - 3}}.