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Chapter 0: Problem 10

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$\frac{x^{2}-8 x+16}{3 x-12}$$

Short Answer

Expert verified
The simplified rational expression is \( \frac{x-4}{3} \) and the excluded value from the domain is \( x=4 \).

Step by step solution

01

Factorize the numerator and denominator

The numerator is a quadratic expression and can be factored as \( (x-4)^2 \), and the denominator is a linear binomial, but has common factors that can be taken out, which leaves \(3(x-4)\). Therefore, the expression becomes: \( \frac{(x-4)^2}{3(x-4)} \).
02

Simplify the Rational Expression

Simplify the rational expression by cancelling the common factors from the numerator and denominator, the (x-4) term can be cancelled out from both the numerator and denominator. This will result in \( \frac{x-4}{3} \) .
03

Find Excluded Values

In order to find the excluded values, set the denominator equal to zero and solve for \(x\). In this case 3(x-4)=0, which gives us x=4. So, x=4 must be excluded from the domain of the rational expression since this value would make the denominator of the original expression zero and thus the expression would be undefined at x=4.

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Most popular questions from this chapter

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