Let's analyze the given problem step-by-step to determine where Julian made his first mistake in factoring the expression [tex]\(2x^4 + 2x^3 - x^2 - x\)[/tex].
Start from the original expression:
[tex]\[ 2x^4 + 2x^3 - x^2 - x \][/tex]
### Step 1
Julian factors out [tex]\(x\)[/tex] from the expression:
[tex]\[ 2x^4 + 2x^3 - x^2 - x = x(2x^3 + 2x^2 - x - 1) \][/tex]
This step is correct.
### Step 2
Julian attempts to further factor the expression inside the parenthesis:
[tex]\[ x(2x^3 + 2x^2 - x - 1) = x\left[2x^2(x + 1) - 1(x - 1)\right] \][/tex]
Here, we should carefully verify the factorized form. Julian's step introduces a mistake because let's see what happens if we expand his factorization:
[tex]\[ 2x^2(x + 1) - 1(x - 1) = 2x^3 + 2x^2 - x + 1 \][/tex]
Comparing this with the original expression inside the parenthesis [tex]\(2x^3 + 2x^2 - x - 1\)[/tex], we see that they don't match since the last term [tex]\(+1\)[/tex] doesn't match [tex]\(-1\)[/tex].
Julian made an error in applying the distributive property when considering [tex]\( -1(x - 1) \)[/tex]. The correct factorization should have been:
[tex]\[ 2x^2(x + 1) - 1(x + 1) = (2x^2 - 1)(x + 1) \][/tex]
### Step 3
Julian proceeds with his mistaken expression:
[tex]\[ x\left(2x^2 - 1\right)(x + 1)(x - 1) \][/tex]
But because the mistake was in Step 2, this step follows the faulty logic from Step 2.
### Conclusion
The first mistake was in Step 2. The correct statement describing this mistake is:
[tex]\[ \text{Statement 2: Julian incorrectly applied the distributive property when factoring out -1} \][/tex]
Thus, Julian's error was in Step 2, and it was due to incorrect application of the distributive property when factoring out [tex]\(-1\)[/tex].
