Select The Correct Step And Statement.Iulian 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 &

Introduction

Factoring expressions is a crucial skill in algebra, and it requires careful attention to detail. In this article, we will analyze Julian's work on factoring the expression $2x^4+2x^3-x^2-x$ and identify the step where he made his first mistake. We will also examine the statement that describes the mistake.

Julian's Work

Julian's work on factoring the expression is shown below:

$${ \begin{array}{r} 2x^4+2x^3-x^2-x \\ \hline 2x^3(x+1)-x(x+1) \\ \hline (2x^3-x)(x+1) \\ \hline \end{array} }$$

Step 1: Identifying the Common Factor

The first step in factoring the expression is to identify the common factor. In this case, the common factor is $(x+1)$.

Step 2: Factoring Out the Common Factor

Julian correctly factored out the common factor $(x+1)$ from the expression $2x^4+2x^3-x^2-x$.

Step 3: Factoring the Remaining Expression

The remaining expression after factoring out the common factor is $2x^3-x$. Julian attempted to factor this expression further.

Step 4: Identifying the Mistake

The mistake occurred in Step 4, where Julian attempted to factor the expression $2x^3-x$ as $(2x^3-x)(x+1)$. However, this is incorrect because the expression $2x^3-x$ cannot be factored further.

Statement Describing the Mistake

The statement that describes the mistake is: "Julian incorrectly factored the expression $2x^3-x$ as $(2x^3-x)(x+1)$."

Conclusion

In conclusion, Julian made his first mistake in Step 4, where he incorrectly factored the expression $2x^3-x$ as $(2x^3-x)(x+1)$. This mistake resulted in an incorrect factorization of the original expression.

Key Takeaways

• Factoring expressions requires careful attention to detail.
• Identifying the common factor is the first step in factoring an expression.
• Factoring out the common factor is a crucial step in factoring an expression.
• The remaining expression after factoring out the common factor may or may not be factorable further.

Common Mistakes in Factoring

• Failing to identify the common factor.
• Incorrectly factoring the remaining expression after factoring out the common factor.
• Not checking the factorization for correctness.

Tips for Factoring

• Carefully examine the expression to identify the common factor.
• Factor out the common factor from the expression.
• Check the factorization for correctness by multiplying the factors together.

Conclusion

Introduction

In our previous article, we analyzed Julian's work on factoring the expression $2x^4+2x^3-x^2-x$ and identified the step where he made his first mistake. We also examined the statement that describes the mistake. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on factoring expressions.

Q&A

Q: What is the first step in factoring an expression?

A: The first step in factoring an expression is to identify the common factor. This involves looking for a factor that is common to all terms in the expression.

Q: How do I identify the common factor?

A: To identify the common factor, look for a term that is common to all terms in the expression. This can be a numerical coefficient, a variable, or a combination of both.

Q: What is the next step after identifying the common factor?

A: After identifying the common factor, the next step is to factor it out from the expression. This involves dividing each term in the expression by the common factor.

Q: Can the remaining expression after factoring out the common factor be factored further?

A: Not always. The remaining expression may or may not be factorable further. It depends on the specific expression and the common factor that was factored out.

Q: What is the most common mistake in factoring expressions?

A: The most common mistake in factoring expressions is failing to identify the common factor. This can lead to incorrect factorization and incorrect solutions to equations.

Q: How can I check my factorization for correctness?

A: To check your factorization for correctness, multiply the factors together and simplify the expression. If the result is the original expression, then your factorization is correct.

Q: What are some tips for factoring expressions?

A: Some tips for factoring expressions include:

• Carefully examine the expression to identify the common factor.
• Factor out the common factor from the expression.
• Check the factorization for correctness by multiplying the factors together.
• Use algebraic properties such as the distributive property to help factor the expression.

Q: Can you provide an example of factoring an expression?

A: Here is an example of factoring an expression:

$$\begin{array}{r} 6x^2+12x+6 \\ \hline 6(x^2+2x+1) \\ \hline 6(x+1)^2 \\ \hline \end{array}$$

In this example, the common factor is 6, and the expression can be factored as $6(x+1)^2$.

Q: What are some common expressions that can be factored?

A: Some common expressions that can be factored include:

• Quadratic expressions in the form of $ax^2+bx+c$
• Cubic expressions in the form of $ax^3+bx^2+cx+d$
• Polynomial expressions in the form of $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$

Q: Can you provide a list of common factoring techniques?

A: Here is a list of common factoring techniques:

• Factoring out the greatest common factor (GCF)
• Factoring by grouping
• Factoring quadratic expressions
• Factoring cubic expressions
• Factoring polynomial expressions

Q: What are some real-world applications of factoring expressions?

A: Factoring expressions has many real-world applications, including:

• Solving systems of equations
• Finding the roots of a polynomial equation
• Analyzing the behavior of a function
• Optimizing a system or process

Conclusion

Factoring expressions is a crucial skill in algebra, and it requires careful attention to detail. By identifying the common factor, factoring out the common factor, and checking the factorization for correctness, we can ensure that our factorization is accurate. We hope that this Q&A section has provided additional information and clarification on factoring expressions.