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

Step 1: Factor the Expression

Firstly, factor both the numerator and the denominator. The numerator \(4x - 8\) can be factored by taking out the common factor of \(4\), which leaves us with \(4(x - 2)\). The denominator is a quadratic which factors to \((x - 2)^{2}\).

Step 2: Simplify the Expression

Now that both the numerator and the denominator are factored, cancel out the same factors from the numerator and the denominator. The term \((x - 2)\) appears in both the numerator and denominator, hence we can cancel it out. This simplifies our expression to \(\frac{4}{x - 2}\).

Step 3: Identify the Excluded Values

A number must be excluded from the domain if it makes the denominator of the simplified expression equal to zero, as division by zero is undefined. Setting \(x - 2 = 0\) and solving for \(x\), we get \(x = 2\). So, \(2\) must be excluded from the domain.