Problem 8 - Algebra and Trigonometry

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 rational expression is \(\frac{4}{x-2}\). And the number that must be excluded from the domain is \(2\).

Step 1: Factorize the numerator and denominator

The numerator, \(4x-8\), can be factorized to \(4(x-2)\). The denominator, \(x^{2}-4x+4\), is a perfect square and can be factorized to \((x-2)^{2}\).

Step 2: Simplify the Expression

Upon factorizing, we have the new expression as \(\frac{4(x-2)}{(x-2)^{2}}\). We can then simplify the expression by canceling out a term of \((x-2)\) in both the numerator and denominator. The simplified solution is \(\frac{4}{x-2}\).

Step 3: Identify the Excluded Values

Recall that a denominator of a fraction cannot be zero. Therefore, the value that makes the denominator zero is excluded from the domain. If \(x-2 = 0\), then \(x = 2\). Hence, \(2\) is excluded from the domain.

Key Concepts

Domain Exclusion

In mathematical terms, the domain of a function or expression includes all the possible values that the variables involved can take. However, rational expressions have certain values that must be excluded from the domain. This is because division by zero is undefined, and any value that makes the denominator zero must therefore be left out.

To determine which numbers are excluded from the domain of a rational expression, follow these steps:

• Set the denominator equal to zero.
• Solve for the variable to find the values that result in zero.

In the given expression, \[\frac{4x-8}{x^2-4x+4}\]setting the denominator \[x^2-4x+4 = 0\]reveals that \(x = 2\) is the value that makes the denominator zero, thus \(x = 2\) must be excluded from the domain.

Factorization

Factorization is the process of breaking down expressions into products of simpler expressions or numbers. It can significantly simplify calculations and reveal useful properties of the expression, such as roots or asymptotes.

For rational expressions, factorization allows us to simplify them by canceling common terms in the numerator and denominator. In the example expression:\[4x-8\text{ and } x^2-4x+4\]First, factorize the numerator:

• Factor out common factors: \(4x - 8 = 4(x - 2)\)

Next, factorize the denominator, which is a perfect square:

• Use quadratic identities: \(x^2 - 4x + 4 = (x - 2)^2\)

This shows the relationship between the numerator and denominator, making simplification easier.

Simplification

Simplification is the process by which an expression is reduced to its simplest form, without changing its value. Simplifying rational expressions involves canceling out common factors from the numerator and the denominator.

In the given expression, after factorization, we have:\[\frac{4(x-2)}{(x-2)^2}\]Since \(x-2\) is a common factor in both, it can be canceled out, resulting in:\[\frac{4}{x-2}\]However, it is crucial to remember that simplification does not affect the domain exclusions derived earlier. So, despite simplifying the expression, the excluded value (\(x = 2\)) remains unchanged.

Denominator Exclusions

When working with rational expressions, one of the key steps is to determine which values to exclude by analyzing the denominator. The denominator exclusions ensure that the expression is defined for the values of the variable.

In any rational expression, if the denominator is set to zero it would make the expression undefined. Therefore, values that make the denominator zero are termed as exclusions.

• Solve the equation given by setting the denominator equal to zero.
• The results of this equation give the excluded values.

For \[\frac{4x-8}{x^2-4x+4}\]the exclusion is \(x=2\) because it makes the denominator zero. In conclusion, always identify these values to ensure your rational expression remains valid.