Question

select the correct step and statement.

julian factored the expression \\(2x^4 + 2x^3 - x^2 - x\\). his work is shown.

at which step did julian make his first mistake, and which statement describes the mistake?

\\(\

$$\begin{array}{|l|l|}\\hline & 2x^4 + 2x^3 - x^2 - x \\\\ \\hline \\text{step 1} & = x(2x^3 + 2x^2 - x - 1) \\\\ \\hline \\text{step 2} & = x2x^2(x + 1) - 1(x - 1) \\\\ \\hline \\text{step 3} & = x(2x^2 - 1)(x + 1)(x - 1) \\\\ \\hline \\end{array}$$

\\)

\\(\

$$\begin{array}{|l|l|}\\hline \\text{statement 1} & \\text{julian should have factored } (2x^2 - 1) \\text{ as a difference of squares.} \\\\ \\hline \\text{statement 2} & \\text{julian incorrectly applied the distributive property when factoring out -1.} \\\\ \\hline \\text{statement 3} & \\text{julian should have factored } 2x \\text{ from all terms instead of } x. \\\\ \\hline \\text{statement 4} & \\text{julian incorrectly factored } 2x^2 \\text{ from the first group of terms.} \\\\ \\hline \\end{array}$$

\\)

Explanation:

Analyze Julian's factorization steps

The original expression is:

$$ 2x^4 + 2x^3 - x^2 - x $$

In Step 1, Julian factors out the greatest common factor \(x\):

$$ x(2x^3 + 2x^2 - x - 1) $$

This step is correct.

In Step 2, Julian attempts to factor by grouping the terms inside the parentheses:

$$ 2x^3 + 2x^2 - x - 1 = 2x^2(x + 1) - 1(x + 1) $$

However, Julian wrote:

$$ x[2x^2(x + 1) - 1(x - 1)] $$

By writing \(-1(x - 1)\), Julian incorrectly factored out \(-1\) from \(-x - 1\). Factoring out \(-1\) from \(-x - 1\) should yield \(-1(x + 1)\). Thus, Step 2 contains the first mistake.

Identify the correct statement

The mistake in Step 2 is that Julian incorrectly factored out \(-1\) from \(-x - 1\), which is an incorrect application of the distributive property:

$$ -1(x - 1) = -x + 1 eq -x - 1 $$

Therefore, Statement 2 correctly describes this mistake: "Julian incorrectly applied the distributive property when factoring out -1."

Answer:

Step

Step 2

Statement

• Statement 1: Julian should have factored \((2x^2 - 1)\) as a difference of squares.
• Statement 2: Julian incorrectly applied the distributive property when factoring out -1. (Correct answer)
• Statement 3: Julian should have factored \(2x\) from all terms instead of \(x\).
• Statement 4: Julian incorrectly factored \(2x^2\) from the first group of terms.