Problem 10 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The simplified rational expression for \(\frac{x^{2} - 8x + 16}{3x - 12}\) is \(\frac{x - 4}{3}\), with \(x = 4\) excluded from the domain.

Step 1: Break Down the Numerator into Its Factors

\((x^{2} - 8x + 16)\) can be factored into \((x- 4)^2\). It was achieved by finding two numbers that multiply to \(16\) and add to \(-8\), which are \(-4\) and \(-4\). So now our function becomes \(\frac{(x - 4)^2}{3x - 12}\).

Step 2: Break Down the Denominator into Its Factors

Identify factors of the denominator \((3x - 12)\). It can be factored by taking out a common factor, which here is \(3\), resulting into \(3(x - 4)\). So now your function looks like \(\frac{(x - 4)^2}{3(x - 4)}\).

Step 3: Simplify the Rational Expression

Simplify by canceling common factors from the numerator and the denominator: \((x - 4)^2\) divided by \(3(x - 4)\) simplifies to \(\frac{x - 4}{3}\).

Step 4: Identify the Excluded Numbers in the Domain

Now identify numbers you should exclude from the domain of the rational expression. This will be the value of x where the original denominator would equal to zero - as division by zero is undefined. From the denominator \(3x - 12 = 0\), we find \(x = 12/3 = 4\). Therefore \(x = 4\) should be excluded from the domain of the simplified rational expression.