Select The Correct Answer.What Is The Factored Form Of $x^3 + 216$?A. $(x-6)\left(x^2+6x+36\right)$ B. $ ( X + 6 ) ( X 2 − 6 X + 36 ) (x+6)\left(x^2-6x+36\right) ( X + 6 ) ( X 2 − 6 X + 36 ) [/tex] C. $(x+6)\left(x^2-12x+36\right)$ D.

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Introduction

Factoring the sum of cubes is a fundamental concept in algebra that allows us to express a polynomial as a product of simpler polynomials. In this article, we will explore the factored form of the expression $x^3 + 216$ and provide a step-by-step guide on how to arrive at the correct answer.

Understanding the Sum of Cubes Formula

The sum of cubes formula states that $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. This formula can be used to factorize expressions of the form $x^3 + c^3$, where $c$ is a constant.

Applying the Sum of Cubes Formula

To factorize the expression $x^3 + 216$, we can use the sum of cubes formula by setting $a = x$ and $b = 6$. This gives us:

x3+216=(x+6)(x2−6x+36)x^3 + 216 = (x + 6)(x^2 - 6x + 36)

Evaluating the Options

Now that we have arrived at the factored form of the expression, let's evaluate the options provided:

  • Option A: $(x-6)\left(x^2+6x+36\right)$
  • Option B: $(x+6)\left(x^2-6x+36\right)$
  • Option C: $(x+6)\left(x^2-12x+36\right)$
  • Option D: (not provided)

Conclusion

Based on our analysis, we can conclude that the correct answer is:

  • Option B: $(x+6)\left(x^2-6x+36\right)$

This is because the factored form of the expression $x^3 + 216$ is indeed $(x+6)\left(x^2-6x+36\right)$, as derived using the sum of cubes formula.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

  1. Step 1: Identify the expression to be factored: $x^3 + 216$
  2. Step 2: Apply the sum of cubes formula by setting $a = x$ and $b = 6$.
  3. Step 3: Substitute the values of $a$ and $b$ into the sum of cubes formula: $(x + 6)(x^2 - 6x + 36)$
  4. Step 4: Evaluate the options provided and select the correct answer.

Tips and Tricks

Here are some tips and tricks to help you master factoring the sum of cubes:

  • Tip 1: Always identify the expression to be factored and apply the sum of cubes formula accordingly.
  • Tip 2: Use the correct values of $a$ and $b$ when substituting into the sum of cubes formula.
  • Tip 3: Evaluate the options provided and select the correct answer.

Common Mistakes

Here are some common mistakes to avoid when factoring the sum of cubes:

  • Mistake 1: Failing to identify the expression to be factored.
  • Mistake 2: Applying the wrong values of $a$ and $b$ when substituting into the sum of cubes formula.
  • Mistake 3: Selecting the incorrect answer.

Real-World Applications

Factoring the sum of cubes has numerous real-world applications in fields such as:

  • Engineering: Factoring the sum of cubes is used to analyze and design complex systems, such as bridges and buildings.
  • Physics: Factoring the sum of cubes is used to describe the motion of objects and predict their behavior under various conditions.
  • Computer Science: Factoring the sum of cubes is used in algorithms and data structures to optimize performance and efficiency.

Conclusion

In conclusion, factoring the sum of cubes is a powerful tool in algebra that allows us to express a polynomial as a product of simpler polynomials. By applying the sum of cubes formula and evaluating the options provided, we can arrive at the correct answer. With practice and patience, you can master factoring the sum of cubes and apply it to real-world problems.

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Introduction

Factoring the sum of cubes is a fundamental concept in algebra that allows us to express a polynomial as a product of simpler polynomials. In this article, we will provide a Q&A guide to help you master factoring the sum of cubes.

Q: What is the sum of cubes formula?

A: The sum of cubes formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

Q: How do I apply the sum of cubes formula?

A: To apply the sum of cubes formula, you need to identify the expression to be factored and set $a = x$ and $b = c$, where $c$ is the constant term.

Q: What are some common mistakes to avoid when factoring the sum of cubes?

A: Some common mistakes to avoid when factoring the sum of cubes include:

  • Failing to identify the expression to be factored.
  • Applying the wrong values of $a$ and $b$ when substituting into the sum of cubes formula.
  • Selecting the incorrect answer.

Q: What are some real-world applications of factoring the sum of cubes?

A: Factoring the sum of cubes has numerous real-world applications in fields such as:

  • Engineering: Factoring the sum of cubes is used to analyze and design complex systems, such as bridges and buildings.
  • Physics: Factoring the sum of cubes is used to describe the motion of objects and predict their behavior under various conditions.
  • Computer Science: Factoring the sum of cubes is used in algorithms and data structures to optimize performance and efficiency.

Q: How do I evaluate the options provided when factoring the sum of cubes?

A: To evaluate the options provided when factoring the sum of cubes, you need to:

  • Identify the expression to be factored.
  • Apply the sum of cubes formula.
  • Substitute the values of $a$ and $b$ into the sum of cubes formula.
  • Evaluate the options provided and select the correct answer.

Q: What are some tips and tricks to help me master factoring the sum of cubes?

A: Some tips and tricks to help you master factoring the sum of cubes include:

  • Always identify the expression to be factored and apply the sum of cubes formula accordingly.
  • Use the correct values of $a$ and $b$ when substituting into the sum of cubes formula.
  • Evaluate the options provided and select the correct answer.

Q: Can you provide a step-by-step solution to a factoring the sum of cubes problem?

A: Here's a step-by-step solution to the problem:

  1. Step 1: Identify the expression to be factored: $x^3 + 216$
  2. Step 2: Apply the sum of cubes formula by setting $a = x$ and $b = 6$.
  3. Step 3: Substitute the values of $a$ and $b$ into the sum of cubes formula: $(x + 6)(x^2 - 6x + 36)$
  4. Step 4: Evaluate the options provided and select the correct answer.

Conclusion

In conclusion, factoring the sum of cubes is a powerful tool in algebra that allows us to express a polynomial as a product of simpler polynomials. By applying the sum of cubes formula and evaluating the options provided, we can arrive at the correct answer. With practice and patience, you can master factoring the sum of cubes and apply it to real-world problems.

Frequently Asked Questions

Here are some frequently asked questions about factoring the sum of cubes:

  • Q: What is the sum of cubes formula? A: The sum of cubes formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
  • Q: How do I apply the sum of cubes formula? A: To apply the sum of cubes formula, you need to identify the expression to be factored and set $a = x$ and $b = c$, where $c$ is the constant term.
  • Q: What are some common mistakes to avoid when factoring the sum of cubes? A: Some common mistakes to avoid when factoring the sum of cubes include:
    • Failing to identify the expression to be factored.
    • Applying the wrong values of $a$ and $b$ when substituting into the sum of cubes formula.
    • Selecting the incorrect answer.

Additional Resources

Here are some additional resources to help you master factoring the sum of cubes:

  • Online Tutorials: Websites such as Khan Academy and Mathway offer interactive tutorials and exercises to help you practice factoring the sum of cubes.
  • Textbooks: Algebra textbooks such as "Algebra and Trigonometry" by Michael Sullivan and "College Algebra" by James Stewart provide comprehensive coverage of factoring the sum of cubes.
  • Practice Problems: Websites such as IXL and Math Open Reference offer practice problems and quizzes to help you test your skills in factoring the sum of cubes.