Factor $x^3-64$.A. $(x+4)\left(x^2-4x+16\right)$ B. $ ( X + 4 ) ( X 2 − 4 X − 16 ) (x+4)\left(x^2-4x-16\right) ( X + 4 ) ( X 2 − 4 X − 16 ) [/tex] C. $(x-4)\left(x^2+4x+16\right)$ D. $(x-4)\left(x^2+4x-16\right)$

Introduction

Factoring is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will focus on factoring the cubic expression $x^3-64$. This expression can be factored using various techniques, including the difference of cubes formula. Our goal is to explore the different methods of factoring and determine the correct factorization of the given expression.

Understanding the Expression

Before we proceed with factoring, let's analyze the given expression $x^3-64$. This expression can be rewritten as $x^3-4^3$, which is a difference of cubes. The difference of cubes formula states that $a^3-b^3=(a-b)(a^2+ab+b^2)$. In this case, $a=x$ and $b=4$.

Applying the Difference of Cubes Formula

Using the difference of cubes formula, we can factor the expression $x^3-64$ as follows:

$x^3-4^3 = (x-4)(x^2+4x+16)$

This is the correct factorization of the given expression. Let's analyze each factor separately.

First Factor: $(x-4)$

The first factor, $(x-4)$, is a linear factor that represents the difference between the variable $x$ and the constant $4$. This factor can be further analyzed by considering its roots. The root of the factor $(x-4)$ is $x=4$, which means that the expression $x^3-64$ equals zero when $x=4$.

Second Factor: $(x^2+4x+16)$

The second factor, $(x^2+4x+16)$, is a quadratic factor that represents a quadratic expression. This factor can be analyzed by considering its roots. However, the quadratic expression $x^2+4x+16$ does not have real roots, as its discriminant is negative. Therefore, the quadratic factor $(x^2+4x+16)$ cannot be factored further using real numbers.

Conclusion

In conclusion, the correct factorization of the cubic expression $x^3-64$ is $(x-4)(x^2+4x+16)$. This factorization can be obtained using the difference of cubes formula, which states that $a^3-b^3=(a-b)(a^2+ab+b^2)$. The first factor, $(x-4)$, represents the difference between the variable $x$ and the constant $4$, while the second factor, $(x^2+4x+16)$, represents a quadratic expression that does not have real roots.

Comparison with Other Options

Let's compare the correct factorization $(x-4)(x^2+4x+16)$ with the other options provided:

• Option A: $(x+4)(x^2-4x+16)$ is incorrect, as the first factor should be $(x-4)$, not $(x+4)$.
• Option B: $(x+4)(x^2-4x-16)$ is incorrect, as the first factor should be $(x-4)$, not $(x+4)$, and the quadratic factor should be $(x^2+4x+16)$, not $(x^2-4x-16)$.
• Option C: $(x-4)(x^2+4x+16)$ is correct, but it is not the only correct option. However, it is the most straightforward and simplest factorization of the given expression.
• Option D: $(x-4)(x^2+4x-16)$ is incorrect, as the quadratic factor should be $(x^2+4x+16)$, not $(x^2+4x-16)$.

Conclusion

In conclusion, the correct factorization of the cubic expression $x^3-64$ is $(x-4)(x^2+4x+16)$. This factorization can be obtained using the difference of cubes formula, which states that $a^3-b^3=(a-b)(a^2+ab+b^2)$. The first factor, $(x-4)$, represents the difference between the variable $x$ and the constant $4$, while the second factor, $(x^2+4x+16)$, represents a quadratic expression that does not have real roots.

Final Answer

The final answer is: $\boxed{(x-4)(x^2+4x+16)}$

Introduction

In our previous article, we explored the factorization of the cubic expression $x^3-64$ using the difference of cubes formula. We also compared the correct factorization with other options provided. In this article, we will address some common questions and concerns related to the factorization of the given expression.

Q: What is the difference of cubes formula?

A: The difference of cubes formula is a mathematical formula that states that $a^3-b^3=(a-b)(a^2+ab+b^2)$. This formula can be used to factorize expressions of the form $a^3-b^3$.

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

A: To apply the difference of cubes formula, you need to identify the values of $a$ and $b$ in the given expression. In the case of $x^3-64$, $a=x$ and $b=4$. Then, you can use the formula to factorize the expression as $(x-4)(x^2+4x+16)$.

Q: What is the significance of the first factor, $(x-4)$?

A: The first factor, $(x-4)$, represents the difference between the variable $x$ and the constant $4$. This factor can be further analyzed by considering its roots. The root of the factor $(x-4)$ is $x=4$, which means that the expression $x^3-64$ equals zero when $x=4$.

Q: What is the significance of the second factor, $(x^2+4x+16)$?

A: The second factor, $(x^2+4x+16)$, represents a quadratic expression that does not have real roots. This factor cannot be factored further using real numbers.

Q: How do I determine the correct factorization of the given expression?

A: To determine the correct factorization of the given expression, you need to apply the difference of cubes formula and compare the result with the other options provided. In this case, the correct factorization is $(x-4)(x^2+4x+16)$.

Q: What are some common mistakes to avoid when factoring the cubic expression $x^3-64$?

A: Some common mistakes to avoid when factoring the cubic expression $x^3-64$ include:

• Using the wrong values of $a$ and $b$ in the difference of cubes formula.
• Not applying the formula correctly.
• Not comparing the result with the other options provided.
• Not considering the significance of the first and second factors.

Q: How can I practice factoring the cubic expression $x^3-64$?

A: You can practice factoring the cubic expression $x^3-64$ by:

• Applying the difference of cubes formula to different values of $a$ and $b$.
• Comparing the result with the other options provided.
• Analyzing the significance of the first and second factors.
• Using online resources or practice problems to reinforce your understanding.

Conclusion

In conclusion, the factorization of the cubic expression $x^3-64$ is a fundamental concept in algebra that involves applying the difference of cubes formula and analyzing the significance of the first and second factors. By understanding the correct factorization and avoiding common mistakes, you can develop a strong foundation in algebra and apply it to a wide range of mathematical problems.

Final Answer

The final answer is: $\boxed{(x-4)(x^2+4x+16)}$