Problem 8 - College Algebra

Question

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$ \frac{4 x-8}{x^{2}-4 x+4} $$

Step-by-Step Solution

Answer

The simplified form of the given rational expression is \(\frac{4}{x - 2}\), with \(x \neq 2\) being the excluded value from the domain.

Step 1: Factor Numerator and Denominator

Factor the numerator and the denominator of the given fraction to find common factors if any. The given expression is \(\frac{4x - 8}{x^{2} - 4x + 4}\). After factoring the expressions we get, \(\frac{4(x - 2)}{(x - 2)(x - 2)}\)

Step 2: Cancel Common Factors

Cancel common factors from the numerator and the denominator to simplify the rational expression. We can simplify \(\frac{4(x - 2)}{(x - 2)(x - 2)}\) as \(\frac{4}{x - 2}\)

Step 3: Determine Excluded Values

Analyze the denominator of the simplified rational expression to find values that would make the denominator zero as these values must be excluded from the domain. In this case, \(x - 2 = 0\), hence \(x = 2\) needs to be excluded from the domain as it would make the denominator zero.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that simplifies expressions and solves equations. At its core, factoring transforms a complex polynomial into a product of simpler polynomials or numbers. For example, the polynomial equation \(x^{2} - 4x + 4\) can be factored into \(x - 2)(x - 2)\), which is also referred to as \(x - 2)^{2}\). Factoring makes it easier to identify and cancel out common factors in rational expressions, leading to further simplification.

To factor, we search for two numbers that multiply to give the last term (in this case, +4) and add to give the coefficient of the middle term (in this case, -4). In our example, \(x^{2} - 4x + 4\), the numbers -2 and -2 fulfill these conditions. Hence, the factored form is \(x - 2)^{2}\). Recognizing patterns, such as the difference of squares or perfect square trinomials, can be very helpful in factoring more complex polynomials effectively.

Excluded Values in Rational Expressions

When dealing with rational expressions, it's imperative to understand the concept of excluded values. These are essentially the values for the variable that would make the denominator of the rational expression equal to zero. Since division by zero is undefined in mathematics, these values must be excluded from the domain of the expression.

In our example, the simplified rational expression is \(\frac{4}{x - 2}\). The excluded value in this case is 2, because if we were to substitute \(x = 2\), the denominator would become zero. It is always important to check for excluded values after simplifying a rational expression, as they are critical to understanding the valid range of the function or expression.

Canceling Common Factors in Rational Expressions

Canceling common factors within rational expressions is a technique used to simplify them to their lowest terms. This process is similar to reducing fractions in basic arithmetic. A rational expression is in simplest form when there are no common factors in the numerator and denominator, other than 1.

The process involves factoring both the numerator and the denominator and then dividing out any factors that appear in both. For instance, in the expression \(\frac{4(x - 2)}{(x - 2)(x - 2)}\), \(x - 2\) is a common factor in both the numerator and the denominator. Upon canceling the common factor, we obtain the simpler expression \(\frac{4}{x - 2}\). This simplification makes it much easier to work with the expression, whether for evaluating, graphing, or integrating into further algebraic operations.