Problem 90 - Elementary Algebra

Question

Reduce the fraction \(\frac{x^{2}-4 x+4}{x^{2}-4}\).

Step-by-Step Solution

Answer

Question: Reduce the given fraction \(\frac{x^{2}-4 x+4}{x^{2}-4}\). Answer: The simplified fraction in its most reduced form is \(\frac{x-2}{x+2}\).

Step 1: Factor the numerator and the denominator

To reduce the fraction, we need to factor both the numerator and the denominator. Let's start with the numerator \((x^{2}-4 x+4)\) and the denominator \((x^{2}-4)\). Numerator: \((x^{2}-4 x+4)\) can be factored as \((x-2)(x-2)\). Denominator: \((x^{2}-4)\) can be factored as \((x-2)(x+2)\) using the difference of squares. So we get the factored form of the given fraction as: \(\frac{(x-2)(x-2)}{(x-2)(x+2)}\).

Step 2: Cancel out the common factors

Now, we can cancel out the common terms in the numerator and the denominator which is \((x-2)\). This simplifies the fraction as: \(\frac{(x-2)}{(x+2)}\).

Step 3: Present the simplified final answer

After canceling out the common terms, the final simplified fraction is: \(\frac{x-2}{x+2}\). This is the most reduced form of the given fraction.

Key Concepts

Fraction Reduction

Reducing fractions is a fundamental concept in algebra, which means making a fraction simpler without changing its value. This process involves finding and canceling out common factors in the numerator and the denominator, which are part of factoring as a whole.
When you reduce a fraction, you make it as simple as possible so that it's easier to understand or use in further calculations. In our original exercise, the fraction \( \frac{x^{2}-4x+4}{x^{2}-4} \) needed reducing.

• We started by factoring both the numerator and denominator.
• After factoring, looked for common factors in both parts of the fraction.
• Once identified, we canceled the common factors, simplifying our fraction.

This revealed the simplest form of our original fraction and ensured all excess components were effectively removed.

Factoring Polynomials

Factoring polynomials is the process of rewriting a polynomial equation as a product of simpler polynomials. A key skill in algebra, this allows us to simplify expressions or solve equations more easily. In the original solution, we needed to factor two specific polynomials: the numerator \( (x^2 - 4x + 4) \) and the denominator \( (x^2 - 4) \).
Factoring the numerator required identifying it as a perfect square trinomial:

• The polynomial \( x^2 - 4x + 4 \) simplifies into \( (x-2)(x-2) \).

For the denominator, we recognize the special pattern of the difference of squares and can factor it accordingly. Understanding these patterns is crucial for efficiently breaking down complex expressions.
Mastering these techniques can dramatically ease the burden of simplifying or solving polynomial equations. It is one of the cornerstones of algebraic manipulation.

Difference of Squares

The difference of squares is a specific pattern that helps in factoring expressions of the form \( a^2 - b^2 \). It is a powerful algebraic tool due to its simplicity and directness.
This pattern states that \( a^2 - b^2 \) can be factored into \( (a-b)(a+b) \).
In the exercise, we see this pattern in the denominator: \( x^2 - 4 \). Here, \( 4 \) is really \( 2^2 \), allowing us to factor it easily as \( (x-2)(x+2) \). Recognizing and applying this concept quickly simplifies the task of factoring certain types of expressions.

• First, identify expressions matching the pattern \( a^2 - b^2 \).
• Apply the difference of squares formula to break down the expression.

The purpose is to simplify the expression by transforming a simple subtraction into a multiplication equation. This can make it easier to identify and cancel common factors in fraction reduction.