Question

multiplying and dividing rational expressions
mathias found the product \\(\frac{4x + 8}{x} \cdot \frac{5}{x + 2}\\) as follows.
mathias’s work
\\(\frac{4x + 8}{x} \cdot \frac{5}{x + 2} = \frac{4x + 8}{x} \cdot \frac{5}{x + 2} = \frac{12}{1} \cdot \frac{5}{x + 2} = \frac{60}{x + 2}\\)
which of the following best describes his error?
he did not divide out the 2 from the 4 and the 8.
he did not distribute the 5 to the terms in the other numerator.
he did not multiply the denominators correctly.
he did not factor the first numerator.

Explanation:

Step1: Factor the first numerator

Factor \(4x + 8\) as \(4(x + 2)\). So the expression becomes \(\frac{4(x + 2)}{x}\cdot\frac{5}{x + 2}\).

Step2: Simplify the expression

Cancel out the common factor \((x + 2)\) from the numerator and the denominator. We get \(\frac{4}{x}\cdot5=\frac{20}{x}\).

Step3: Analyze Mathias's error

Mathias did not factor the first numerator \(4x + 8\) correctly. He incorrectly simplified \(4x+8\) to \(12\) instead of factoring it as \(4(x + 2)\). So the error is that he did not factor the first numerator.

Answer:

He did not factor the first numerator.