Select The Correct Step And Statement.Julian Factored The Expression $2x^4 + 2x^3 - X^2 - X$. His Work Is Shown Below. At Which Step Did Julian Make His First Mistake, And Which Statement Describes The

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Julian's Factoring Work

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Julian factored the expression $2x^4 + 2x^3 - x^2 - x$ using the following steps:

1. Step 1: $2x^4 + 2x^3 - x^2 - x = 2x^3(x + 1) - x(x + 1)$
2. Step 2: $2x^3(x + 1) - x(x + 1) = (2x^3 - x)(x + 1)$
3. Step 3: $(2x^3 - x)(x + 1) = 2x^3(x + 1) - x(x + 1)$
4. Step 4: $2x^3(x + 1) - x(x + 1) = (2x^3 - x)(x + 1)$

Statement Analysis

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The following statements describe Julian's work:

• Statement 1: Julian made his first mistake at step 2.
• Statement 2: Julian made his first mistake at step 3.
• Statement 3: Julian made his first mistake at step 4.
• Statement 4: Julian made his first mistake at step 1.

Correct Answer

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To determine the correct answer, we need to analyze each step and statement.

Step 1 Analysis

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In step 1, Julian correctly factored out the common binomial factor $(x + 1)$ from the first two terms and the last two terms.

Step 2 Analysis

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In step 2, Julian incorrectly wrote the expression as $(2x^3 - x)(x + 1)$. This is incorrect because the correct factorization is $2x^3(x + 1) - x(x + 1)$.

Step 3 Analysis

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In step 3, Julian repeated the same mistake as in step 2, writing the expression as $(2x^3 - x)(x + 1)$.

Step 4 Analysis

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In step 4, Julian repeated the same mistake as in steps 2 and 3, writing the expression as $(2x^3 - x)(x + 1)$.

Conclusion

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Based on the analysis, Julian made his first mistake at step 2. The correct factorization is $2x^3(x + 1) - x(x + 1)$, not $(2x^3 - x)(x + 1)$.

Correct Factorization

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The correct factorization of the expression $2x^4 + 2x^3 - x^2 - x$ is:

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

However, this is not the final answer. We need to factor out the common binomial factor $(x + 1)$ from the first two terms and the last two terms.

Final Answer

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The final answer is:

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

Explanation

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To factor out the common binomial factor $(x + 1)$ from the first two terms and the last two terms, we need to multiply the first two terms by $-1$ and then add the two expressions.

Step-by-Step Solution

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Here is the step-by-step solution:

1. Step 1: $2x^4 + 2x^3 - x^2 - x = 2x^3(x + 1) - x(x + 1)$
2. Step 2: $2x^3(x + 1) - x(x + 1) = (2x^3 - x)(x + 1)$
3. Step 3: $(2x^3 - x)(x + 1) = 2x^3(x + 1) - x(x + 1)$
4. Step 4: $2x^3(x + 1) - x(x + 1) = (2x^3 - x)(x + 1)$
5. Step 5: $(2x^3 - x)(x + 1) = x(2x^2 + 1)(x + 1)$

Conclusion

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In conclusion, Julian made his first mistake at step 2. The correct factorization of the expression $2x^4 + 2x^3 - x^2 - x$ is $x(2x^2 + 1)(x + 1)$.

Final Answer

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The final answer is $x(2x^2 + 1)(x + 1)$.

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Understanding Julian's Mistake

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In the previous article, we analyzed Julian's work and determined that he made his first mistake at step 2. However, we also realized that the correct factorization of the expression $2x^4 + 2x^3 - x^2 - x$ is not $(2x^3 - x)(x + 1)$, but rather $x(2x^2 + 1)(x + 1)$.

Q&A Session

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Here are some frequently asked questions and answers related to Julian's mistake:

Q: What was Julian's mistake?

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A: Julian's mistake was writing the expression as $(2x^3 - x)(x + 1)$ instead of $2x^3(x + 1) - x(x + 1)$.

Q: Why is the correct factorization $x(2x^2 + 1)(x + 1)$?

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A: The correct factorization is $x(2x^2 + 1)(x + 1)$ because we need to factor out the common binomial factor $(x + 1)$ from the first two terms and the last two terms.

Q: What is the final answer?

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A: The final answer is $x(2x^2 + 1)(x + 1)$.

Q: Why did Julian make his mistake?

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A: Julian made his mistake because he incorrectly wrote the expression as $(2x^3 - x)(x + 1)$ instead of $2x^3(x + 1) - x(x + 1)$.

Q: How can we avoid making the same mistake as Julian?

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A: To avoid making the same mistake as Julian, we need to carefully analyze each step and make sure that we are writing the correct expression.

Common Mistakes

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Here are some common mistakes that students make when factoring expressions:

• Mistake 1: Writing the expression as $(2x^3 - x)(x + 1)$ instead of $2x^3(x + 1) - x(x + 1)$.
• Mistake 2: Failing to factor out the common binomial factor $(x + 1)$ from the first two terms and the last two terms.
• Mistake 3: Writing the expression as $(2x^3 - x)(x + 1)$ instead of $x(2x^2 + 1)(x + 1)$.

Tips for Factoring Expressions

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Here are some tips for factoring expressions:

• Tip 1: Make sure to carefully analyze each step and make sure that you are writing the correct expression.
• Tip 2: Factor out the common binomial factor from the first two terms and the last two terms.
• Tip 3: Use the distributive property to expand the expression and then factor out the common binomial factor.

Conclusion

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In conclusion, Julian made his first mistake at step 2. The correct factorization of the expression $2x^4 + 2x^3 - x^2 - x$ is $x(2x^2 + 1)(x + 1)$. We also discussed some common mistakes that students make when factoring expressions and provided some tips for factoring expressions.

Final Answer

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The final answer is $x(2x^2 + 1)(x + 1)$.