Problem 39 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{4 x^{2}-4}{4 x^{2}+4}$$

Step-by-Step Solution

Answer

\(\frac{(x - 1)(x + 1)}{x^2 + 1}\)

Step 1: Identify common factors in the numerator

Look for common factors in the numerator. The numerator is \(4x^2 - 4\). Notice that both terms have a common factor of 4: \[4x^2 - 4 = 4(x^2 - 1)\]

Step 2: Identify common factors in the denominator

Similarly, look for common factors in the denominator. The denominator is \(4x^2 + 4\). Both terms have a common factor of 4: \[4x^2 + 4 = 4(x^2 + 1)\]

Step 3: Simplify the rational expression

With the expressions factored, the rational expression becomes: \[\frac{4(x^2 - 1)}{4(x^2 + 1)}\] Now, cancel the common factor of 4 from both the numerator and denominator: \[\frac{x^2 - 1}{x^2 + 1}\]

Step 4: Factorize the remaining expression if possible

Check if the remaining expression can be further simplified. The numerator \(x^2 - 1\) can be factored as \((x - 1)(x + 1)\). However, \(x^2 + 1\) does not factorize further with real numbers. Therefore, the simplified expression is: \[\frac{(x - 1)(x + 1)}{x^2 + 1}\]

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in simplifying rational expressions. It involves breaking down a polynomial into simpler polynomials that can be multiplied together to obtain the original polynomial. For example, consider the polynomial in the numerator of our expression, \(4x^2 - 4\). By identifying the greatest common factor (GCF), which in this case is 4, we can factor the polynomial as follows:
\(4x^2 - 4 = 4(x^2 - 1)\).
Similarly, for the denominator \(4x^2 + 4\), the GCF is also 4, allowing us to factor it into \(4(x^2 + 1)\).

Understanding how to factor polynomials helps in simplifying rational expressions and making further reductions possible. So, always look for the GCF and possible factorization patterns like difference of squares, which brings us to another common factorization: \(x^2 - 1 = (x - 1)(x + 1)\).

Numerator and Denominator

The terms 'numerator' and 'denominator' are crucial in understanding rational expressions. The numerator is the top part of the fraction, and the denominator is the bottom part. In our expression, \(\frac{4x^2 - 4}{4x^2 + 4}\),
the numerator is \(4x^2 - 4\) and the denominator is \(4x^2 + 4\).

Factoring polynomials within both parts can simplify the entire rational expression. In our example, both the numerator and denominator have a common factor of 4. By factoring out this common factor, we get:
\(\frac{4(x^2 - 1)}{4(x^2 + 1)}\).

Once the expression is factored, you can cancel the common factors that appear in both the numerator and the denominator, simplifying the expression to \(\frac{x^2 - 1}{x^2 + 1}\).
This form is much simpler and easier to work with.

Common Factors

Identifying common factors is a key step in simplifying rational expressions. A common factor is a number or expression that divides two or more numbers or terms without a remainder. In our exercise, the common factor between the numerator \(4x^2 - 4\) and the denominator \(4x^2 + 4\) is 4.

By factoring out this common factor from both the numerator and the denominator, the expression simplifies significantly:
\(\frac{4(x^2 - 1)}{4(x^2 + 1)}\).
Cancelling out the common factor of 4 leaves us with \(\frac{x^2 - 1}{x^2 + 1}\).

Sometimes, after cancelling out common factors, you may still be able to factorize further. For instance, \(x^2 - 1\) can be further broken down to \((x - 1)(x + 1)\). But always check if further factorization works within the given constraints. In this case, \(x^2 + 1\) remains as it is because it cannot be further factored into real numbers. Understanding and identifying common factors help in simplifying complex rational expressions for better clarity and ease of solving.