Question

1. check for reasonableness a student is asked to factor $r^{10}+r^{8}$ completely. is the students work correct? explain.

$r^{10}+r^{8}$
$\

$$\begin{array}{r}10 \\\\ \\textcircled{2} 5\\end{array}$$

$ $\

$$\begin{array}{r}8 \\\\ \\textcircled{2} 4\\end{array}$$

$
$r^{2}(r^{5}+r^{4})$

Explanation:

Step1: Factor out the greatest common factor

The greatest common factor of \(r^{10}\) and \(r^{8}\) is \(r^{8}\), not \(r^{2}\). Using the rule \(a^{m}\div a^{n}=a^{m - n}\), when factoring \(r^{10}+r^{8}\), we have \(r^{10}+r^{8}=r^{8}(r^{10 - 8}+r^{8 - 8})=r^{8}(r^{2}+1)\).

Step2: Analyze the student's work

The student factored out \(r^{2}\) and got \(r^{2}(r^{5}+r^{4})\). But \(r^{5}+r^{4}\) can be factored further as \(r^{4}(r + 1)\). However, the initial step of factoring out the wrong common factor (\(r^{2}\) instead of \(r^{8}\)) makes the student's work incorrect.

Answer:

The student's work is incorrect. The correct factoring is \(r^{8}(r^{2}+1)\) as the greatest - common factor of \(r^{10}\) and \(r^{8}\) is \(r^{8}\), not \(r^{2}\).