Question

Perform the indicated operation and simplify the result. Leave your answer in factored form. $$ \frac{\frac{x-2}{4 x}}{\frac{x^{2}-4 x+4}{12 x}} $$

Short answer

\( \frac{3}{x-2} \)

Step-by-step solution

Simplify the Numerator

Simplify the numerator of the main fraction. The numerator is \( \frac{x-2}{4x} \).

Simplify the Denominator

Simplify the denominator of the main fraction. The denominator is \( \frac{x^{2}-4x+4}{12x} \). Notice that \(x^2 - 4x + 4 = (x-2)^2\). So the denominator can be rewritten as \( \frac{(x-2)^2}{12x} \).

Divide by the Reciprocal

Divide by the reciprocal of the denominator. This means turning the division into multiplication by flipping the denominator fraction: \[ \frac{x-2}{4x} \div \frac{(x-2)^2}{12x} = \frac{x-2}{4x} \times \frac{12x}{(x-2)^2} \].

Simplify the Result

Cancel out common terms in the multiplication: \[ \frac{x-2}{4x} \times \frac{12x}{(x-2)^2} = \frac{12x(x-2)}{4x(x-2)(x-2)} \], this simplifies to \[ \frac{12}{4(x-2)} = \frac{3}{x-2} \].

Key concepts

Factoring Polynomials

Factoring polynomials is a crucial step in simplifying rational expressions. It involves expressing a polynomial as a product of its factors. For instance, in the exercise, the polynomial in the denominator can be factored as \((x^2 - 4x + 4 = (x-2)^2)\). This simplification is achieved by recognizing that \((x^2 - 4x + 4)\) is a perfect square trinomial. Factoring polynomials allows us to cancel out common terms later in the simplification process, making the entire expression easier to handle.

Reciprocal

The concept of the reciprocal is vital when dividing fractions. The reciprocal of a fraction simply involves flipping its numerator and denominator. For example, the reciprocal of \(\frac{3}{4}\) is \ \frac{4}{3} \. In our exercise, to divide by the denominator \ \frac{(x-2)^2}{12x} \, we multiply by its reciprocal, \ \frac{12x}{(x-2)^2} \. This step converts a division problem into a multiplication problem, which is much easier to solve. Understanding and effectively using reciprocal values is a key skill in simplifying complex rational expressions.

Simplification of Fractions

Simplifying fractions involves reducing them to their simplest form by canceling out common factors in the numerator and the denominator. In the provided example, after multiplying by the reciprocal, we get \ \frac{(x-2)}{4x} \times \frac{12x}{(x-2)^2} \. Notice the common \(x-2\) and \(x\) terms in both the numerator and the denominator. Canceling these terms simplifies the expression to \ \frac{12}{4(x-2)}\, which further simplifies to \ \frac{3}{x-2} \. Always look for common factors when simplifying fractions to make calculations more straightforward.

Polynomial Division

Polynomial division is used when dividing one polynomial by another. It is a step-by-step process that reduces these complex expressions into simpler forms. In the exercise, we initially divided one fraction by another. To perform this division, we convert it to a multiplication problem using the reciprocal. After reciprocal transformation, polynomial division converts into a simpler multiplication problem where we can easily cancel common terms. This step is essential as it enables us to transform complicated expressions into manageable ones, simplifying our calculations and solution.