Let's analyze Julian's work step by step to identify where the first mistake occurs.
1. Given Expression:
[tex]\[
2 x^4 + 2 x^3 - x^2 - x
\][/tex]
2. Step 1:
[tex]\[
= x\left(2 x^3 + 2 x^2 - x - 1\right)
\][/tex]
3. Step 2:
[tex]\[
= x\left[2 x^2(x + 1) - 1(x - 1)\right]
\][/tex]
Here, Julian attempts to factor by grouping. He separates the terms into two groups:
[tex]\[
2 x^3 + 2 x^2 \quad \text{and} \quad - x - 1
\][/tex]
He then factors [tex]\(2 x^2\)[/tex] from the first group and [tex]\(-1\)[/tex] from the second group:
[tex]\[
= x \left[2 x^2(x + 1) - 1(x - 1)\right]
\][/tex]
Upon examining this step, we can see that the factoring is done incorrectly.
- For the first part of the expression, Julian correctly factors [tex]\(2 x^2\)[/tex] out of [tex]\(2 x^3 + 2 x^2\)[/tex], resulting in [tex]\(2 x^2 (x + 1)\)[/tex].
- For the second part, when factoring out [tex]\(-1\)[/tex] from [tex]\(- x - 1\)[/tex], it should have been [tex]\(-1 (x + 1)\)[/tex] (factoring [tex]\(-1\)[/tex] correctly here).
Therefore, Julian's factoring of the second term [tex]\(-x - 1\)[/tex] to [tex]\(-1 (x - 1)\)[/tex] was incorrect. It should have been:
[tex]\[
-1 (x + 1)
\][/tex]
4. Step 3 (Julian's Incorrect Work):
[tex]\[
= x\left(2 x^2 - 1\right)(x + 1)(x - 1)
\][/tex]
Since the step that includes the error is Step 2, the correct statement that describes Julian’s mistake is:
Statement 4: Julian incorrectly factored [tex]\(2 x^2\)[/tex] from the first group of terms.
Thus, Julian made the first mistake at Step 2.
