Question

Mark multiplied the expression
2
x
3 −27
6x+30
2
x
⋅
+9x+20
2
x
−x−12
incorrectly.
First, Mark factored each of the polynomial expressions:
3(x+3)(x−3)
6(x+5)
Next, Mark divided out common factors and got the answer: .
x+3
2
Determine Mark’s error.
Option #1: Mark factored incorrectly.
Option #2: Mark divided out common factors incorrectly.
Option #3: Mark did not fully simplify the expression.
(1 point)
Option # shows Mark’s error

Answer 1

To determine where Mark went wrong, we need to analyze the factorization and the cancellation step he took.

1. The first expression that Mark factored is \(2x^3 - 27\). Using the difference of cubes formula, \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\):
- Here, \(2x^3\) can be rewritten as \((\sqrt[3]{2}x)^3\) and \(27\) as \((3)^3\). Thus, it factors to:
\[
( \sqrt[3]{2}x - 3 ) \left( (\sqrt[3]{2}x)^2 + 3(\sqrt[3]{2}x)3 + 3^2 \right)
\]

This is a more complex factorization than Mark presented, indicating he likely did not factor \(2x^3 - 27\) correctly as \(3(x + 3)(x - 3)\).

2. The second expression is \(6x + 30\), which can be factored out as \(6(x + 5)\), and this part seems to be factored correctly by Mark.

3. Mark did not transfer the entire product correctly into the next steps.

4. Regarding the last expression \(2x^2 - x - 12\), it factors into \( (2x + 3)(x - 4) \). Mark's overall simplification did not consider these factors correctly.

Then, Mark successfully divided out common factors but didn't do so after fully simplifying the full expression correctly.

Given the above analysis, we can conclude that Mark's error lies primarily in the first stage of factoring \(2x^3 - 27\) improperly, thus the correct option is:

**Option #1: Mark factored incorrectly.**