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Chapter 0: Problem 8

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

Short Answer

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

Step by step solution

01

Factorize the Expression

To simplify the rational expression, it is factored. Starting with the numerator 4x-8, the common factor which is 4 is factored out. So, 4x-8 is factored to 4(x-2). The denominator is a quadratic expression x^{2}-4x+4. This is a perfect-square trinomial which is the square of (x-2). So, \(x^2 - 4x + 4\) is factored to \( (x-2)^2\). Applying these to the rational expression, \(\frac{4x-8}{x^{2}-4x+4}\) is simplified to \(\frac{4(x-2)}{(x-2)^2}\).
02

Simplify the Expression

After factorizing, the next task is to simplify the expression. The factor (x-2) in both the numerator and denominator is cancelled out, thus making the expression \(\frac{4}{x-2}\).
03

Find the Domain

For the rational expression to be valid, the denominator must not be equal to zero. Thus, the value of x that makes (x-2)=0 is excluded from the domain. Solving \(x-2 = 0\), we get \(x = 2\). Hence, 2 must be excluded from the domain.

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Most popular questions from this chapter

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