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Julian factored the expression [tex]\(2x^4 + 2x^3 - x^2 - x\)[/tex]. His work is shown below. At which step did Julian make his first mistake, and which statement describes the mistake?

\begin{tabular}{|l|c|}
\hline
& [tex]\(2x^4 + 2x^3 - x^2 - x\)[/tex] \\
\hline
Step 1 & [tex]\(= x(2x^3 + 2x^2 - x - 1)\)[/tex] \\
\hline
Step 2 & [tex]\(= x[2x^2(x + 1) - 1(x - 1)]\)[/tex] \\
\hline
Step 3 & [tex]\(= x(2x^2 - 1)(x + 1)(x - 1)\)[/tex] \\
\hline
\end{tabular}

\begin{tabular}{|l|l|}
\hline
Statement 1 & Julian should have factored [tex]\((2x^2 - 1)\)[/tex] as a difference of squares. \\
\hline
Statement 2 & Julian incorrectly applied the distributive property when factoring out [tex]\(-1\)[/tex]. \\
\hline
Statement 3 & Julian should have factored [tex]\(2x\)[/tex] from all terms instead of [tex]\(x\)[/tex]. \\
\hline
Statement 4 & Julian incorrectly factored [tex]\(2x^2\)[/tex] from the first group of terms. \\
\hline
\end{tabular}


Sagot :

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.