Question

Perform the indicated operation and simplify the result. Leave your answer in factored form. $$ \frac{\frac{4-x}{4+x}}{\frac{4 x}{x^{2}-16}} $$

Short answer

-\frac{(x-4)^2}{4x}

Step-by-step solution

- Rewrite the complex fraction

Rewrite the given complex fraction as a division problem: \[ \frac{\frac{4-x}{4+x}}{\frac{4x}{x^2-16}} = \frac{4-x}{4+x} \times \frac{x^2-16}{4x} \]

- Factorize the denominators and numerators

Factorize the terms where possible: \[ x^2-16 = (x-4)(x+4) \] So the expression becomes: \[ \frac{4-x}{4+x} \times \frac{(x-4)(x+4)}{4x} \]

- Simplify the complex fraction

Remember that \(4-x = -(x-4)\). Replace \(4-x\) with \(-(x-4)\): \[ \frac{-(x-4)}{4+x} \times \frac{(x-4)(x+4)}{4x} \]

- Cancel out common factors

Cancel out the common factors in the numerator and denominator. The \((x+4)\) terms cancel out: \[ \frac{-(x-4)}{4+x} \times \frac{(x-4)}{4x} = \frac{-(x-4)^2}{4x} \]

- Simplify the final expression

Combine like terms and simplify the fraction if possible: \[ \frac{-(x-4)^2}{4x} \]

Key concepts

Factoring

Factoring is a key concept in algebra that involves breaking down expressions into simpler components. This is especially useful in simplifying complex fractions.
In our exercise, we need to factor the quadratic expression in the denominator:

• The expression is: \[ x^2 - 16 \]
• Recognize that this is a difference of squares which factors into: \[ (x-4)(x+4) \]

Understanding how to factor expressions is critical as it allows you to cancel out common terms later on when simplifying complex fractions. In this example, we used factoring to break down \[ x^2 - 16 \] into its simpler components \[ (x-4)(x+4) \]. This step is foundational to simplifying the given problem.

Simplification

Simplification refers to making an expression easier to work with by reducing it to its simplest form. In this exercise, we worked on simplifying a complex fraction step by step.
First, we rewrote the complex fraction as a division problem:

• Original: \[ \frac{\frac{4-x}{4+x}}{\frac{4x}{x^2-16}} \]
• Rewritten: \[ \frac{4-x}{4+x} \times \frac{x^2-16}{4x} \]

Next, we factored the quadratic expression as: \[ x^2-16 = (x-4)(x+4) \]. This allowed us to cancel out common factors in the numerator and denominator.
Finally, we remembered that: \[ 4 - x = -(x - 4) \], transformed our expression, canceled common factors, and simplified it to get: \[ \frac{-(x-4)^2}{4x} \].
Each step makes the expression simpler and easier to manage, demonstrating the power of simplification in algebra. Breaking problems into smaller, more manageable parts is a useful strategy in many areas of math.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. Working with these expressions often involves manipulating them to find a simpler or more useful form.
In this exercise, we started with a complex fraction that we needed to simplify by performing indicated operations and rewriting components:

• Numerator: \( \frac{4-x}{4+x} \)
• Denominator: \( \frac{4x}{x^2-16} \)

Factoring, recognizing equivalent forms, and canceling common factors are all techniques used with algebraic expressions.
By applying these techniques, we transformed our initial complex fraction into the simpler expression:\[ \frac{-(x-4)^2}{4x} \]. This shows how manipulating algebraic expressions isn't just about following steps blindly but understanding the structures and relationships within the expressions.
Grasping these underlying principles can make solving problems more intuitive and less daunting. Hence, learning how to work with algebraic expressions is crucial for mastering algebra and higher-level math.