Equivalent Expression: Simplify (4x^2 + 16x - 48) / (x^2 - 36)

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Let's break down how to find an equivalent expression for the given rational expression. This involves factoring, simplifying, and understanding the restrictions on the variable x. We'll go through each step to make it crystal clear.

Understanding the Problem

We're given the expression:

4x2+16x−48x2−36\frac{4x^2 + 16x - 48}{x^2 - 36}

Our goal is to simplify this expression into one of the given options. The conditions x ≠ -6 and x ≠ 6 are crucial because they tell us the values of x that would make the denominator zero, which is undefined in mathematics. These restrictions will remain important throughout our simplification process.

Step-by-Step Simplification

Let's simplify the given expression step by step. This involves factoring both the numerator and the denominator and then canceling out any common factors.

1. Factor the Numerator

The numerator is a quadratic expression: 4x² + 16x - 48. To factor this, we first look for a common factor among all the terms. In this case, each term is divisible by 4. Factoring out the 4, we get:

4(x² + 4x - 12)

Now, we need to factor the quadratic expression inside the parentheses, x² + 4x - 12. We are looking for two numbers that multiply to -12 and add up to 4. Those numbers are 6 and -2. Therefore, we can factor the quadratic as:

(x + 6)(x - 2)

So, the fully factored numerator is:

4(x + 6)(x - 2)

2. Factor the Denominator

The denominator is x² - 36. This is a difference of squares, which can be factored as:

(x + 6)(x - 6)

3. Simplify the Rational Expression

Now we have the factored expression:

4(x+6)(x−2)(x+6)(x−6)\frac{4(x + 6)(x - 2)}{(x + 6)(x - 6)}

We can cancel out the common factor of (x + 6) from both the numerator and the denominator, keeping in mind that x ≠ -6.

This leaves us with:

4(x−2)(x−6)\frac{4(x - 2)}{(x - 6)}

Which simplifies to:

4x−8x−6\frac{4x - 8}{x - 6}

Matching the Simplified Expression with the Options

Now, let's compare our simplified expression with the given options:

A. $\frac{4x - 8}{x - 6}$ B. $\frac{4x + 8}{x + 6}$ C. $\frac{x^2 + 4x - 12}{x - 6}$

Our simplified expression matches option A.

Conclusion

Therefore, the equivalent expression to $\frac{4x^2 + 16x - 48}{x^2 - 36}$, where x ≠ -6 and x ≠ 6, is:

4x−8x−6\frac{4x - 8}{x - 6}

So, the correct answer is A. Factoring and simplifying rational expressions is a fundamental skill in algebra. Remember to always look for common factors and apply factoring techniques such as difference of squares or quadratic factoring. Always keep the restrictions on the variable in mind to avoid undefined expressions. Understanding these steps helps in solving similar problems efficiently and accurately.

Additional Insights for Polynomial Expressions

When dealing with polynomial expressions, especially rational ones, several key concepts come into play. These concepts, when mastered, allow for efficient and accurate simplification and manipulation of complex algebraic expressions. Here are some additional insights that can enhance your understanding and skills.

Importance of Factoring

Factoring is the backbone of simplifying rational expressions. It involves breaking down a polynomial into its constituent factors. Different techniques apply to different types of polynomials:

  • Common Factoring: Always look for common factors first. This simplifies the expression and makes further factoring easier. For example, in the expression 6x² + 12x, the common factor is 6x, so the expression becomes 6x(x + 2).
  • Difference of Squares: Expressions in the form a² - b² can be factored as (a + b) (a - b). This is a straightforward pattern to recognize and apply.
  • Quadratic Factoring: Quadratic expressions ax² + bx + c can be factored into the form (px + q) (rx + s), where p, q, r, and s are constants. This often involves trial and error or using the quadratic formula to find the roots.
  • Sum and Difference of Cubes: These have specific factoring patterns:
    • a³ + b³ = (a + b) (a² - ab + b²)
    • a³ - b³ = (a - b) (a² + ab + b²)

Restrictions on Variables

Rational expressions have restrictions on the variables because division by zero is undefined. Always identify these restrictions before simplifying the expression. For example, in the expression $\frac{1}{x - 3}$, x cannot be 3. These restrictions must be carried through the entire simplification process.

Simplifying Rational Expressions

Simplifying rational expressions involves factoring both the numerator and the denominator and then canceling out common factors. Here are some tips:

  • Factor Completely: Ensure that both the numerator and the denominator are fully factored before attempting to cancel any terms.
  • Cancel Common Factors: Only factors that are common to both the numerator and the denominator can be canceled.
  • State Restrictions: Always state any restrictions on the variable after simplifying the expression.

Combining Rational Expressions

To add or subtract rational expressions, they must have a common denominator. Here are the steps:

  • Find the Least Common Denominator (LCD): This is the smallest expression that is divisible by both denominators.
  • Rewrite Each Fraction: Rewrite each fraction with the LCD as the denominator.
  • Combine the Numerators: Add or subtract the numerators, keeping the common denominator.
  • Simplify: Simplify the resulting expression by factoring and canceling common factors.

Example: Combining Rational Expressions

Let's combine the following expressions:

1x+1+2x−1\frac{1}{x + 1} + \frac{2}{x - 1}

  1. Find the LCD: The LCD is (x + 1) (x - 1).
  2. Rewrite Each Fraction:$\frac{1}{x + 1} \cdot \frac{x - 1}{x - 1} + \frac{2}{x - 1} \cdot \frac{x + 1}{x + 1} = \frac{x - 1}{(x + 1)(x - 1)} + \frac{2(x + 1)}{(x + 1)(x - 1)}$
  3. Combine the Numerators:$\frac{x - 1 + 2x + 2}{(x + 1)(x - 1)} = \frac{3x + 1}{(x + 1)(x - 1)}$
  4. Simplify: The expression is already simplified, and the restrictions are x ≠ -1 and x ≠ 1.

Practice Problems

To solidify your understanding, try these practice problems:

  1. Simplify: $\frac{x^2 - 4}{x^2 + 4x + 4}$
  2. Combine: $\frac{3}{x - 2} - \frac{1}{x + 2}$

By mastering these concepts and practicing regularly, you can become proficient in simplifying and manipulating polynomial expressions. This is a critical skill for advanced algebra and calculus.

In conclusion, simplifying rational expressions involves factoring, identifying restrictions, and canceling common factors. Mastering these steps leads to efficient and accurate solutions.

For further learning, you can visit Khan Academy's Algebra Section for more practice and resources.