What Is The Completely Factored Form Of $x^3 - 64x$?$A. $x(x-8)(x-8)$B. $(x-4)\left(x^2+4x+16\right)$C. \$x(x-8)(x+8)$[/tex\]D. $(x-4)(x+4)(x+4)$

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Introduction

Factoring polynomials is a fundamental concept in algebra, and it plays a crucial role in solving equations and inequalities. In this article, we will explore the completely factored form of the polynomial x3−64xx^3 - 64x. We will examine each option and determine which one is the correct completely factored form.

Understanding the Polynomial

Before we dive into factoring, let's take a closer look at the polynomial x3−64xx^3 - 64x. This polynomial can be rewritten as x(x2−64)x(x^2 - 64). We can further simplify the expression by recognizing that 6464 is a perfect square, which is 828^2. Therefore, we can rewrite the polynomial as x(x−8)(x+8)x(x - 8)(x + 8).

Option A: x(x−8)(x−8)x(x-8)(x-8)

Option A is x(x−8)(x−8)x(x-8)(x-8). At first glance, this option may seem plausible, but it is not the correct completely factored form. The reason is that the polynomial x3−64xx^3 - 64x has a repeated factor of xx, but it does not have a repeated factor of (x−8)(x-8). In other words, the polynomial can be factored as x(x−8)(x+8)x(x-8)(x+8), but not as x(x−8)(x−8)x(x-8)(x-8).

Option B: (x−4)(x2+4x+16)(x-4)\left(x^2+4x+16\right)

Option B is (x−4)(x2+4x+16)(x-4)\left(x^2+4x+16\right). This option is also not the correct completely factored form. The reason is that the polynomial x3−64xx^3 - 64x does not have a factor of (x−4)(x-4), and the quadratic expression x2+4x+16x^2+4x+16 is not a factor of the polynomial.

Option C: x(x−8)(x+8)x(x-8)(x+8)

Option C is x(x−8)(x+8)x(x-8)(x+8). This option is the correct completely factored form of the polynomial x3−64xx^3 - 64x. The reason is that the polynomial can be factored as x(x−8)(x+8)x(x-8)(x+8), which is a product of three binomials.

Option D: (x−4)(x+4)(x+4)(x-4)(x+4)(x+4)

Option D is (x−4)(x+4)(x+4)(x-4)(x+4)(x+4). This option is not the correct completely factored form. The reason is that the polynomial x3−64xx^3 - 64x does not have a factor of (x−4)(x-4), and the quadratic expression (x+4)(x+4)(x+4)(x+4) is not a factor of the polynomial.

Conclusion

In conclusion, the completely factored form of the polynomial x3−64xx^3 - 64x is x(x−8)(x+8)x(x-8)(x+8). This option is the correct answer because it is a product of three binomials, and it accurately represents the polynomial.

Frequently Asked Questions

  • What is the completely factored form of x3−64xx^3 - 64x? The completely factored form of x3−64xx^3 - 64x is x(x−8)(x+8)x(x-8)(x+8).
  • Why is option A not the correct completely factored form? Option A is not the correct completely factored form because the polynomial x3−64xx^3 - 64x does not have a repeated factor of (x−8)(x-8).
  • Why is option B not the correct completely factored form? Option B is not the correct completely factored form because the polynomial x3−64xx^3 - 64x does not have a factor of (x−4)(x-4), and the quadratic expression x2+4x+16x^2+4x+16 is not a factor of the polynomial.
  • Why is option D not the correct completely factored form? Option D is not the correct completely factored form because the polynomial x3−64xx^3 - 64x does not have a factor of (x−4)(x-4), and the quadratic expression (x+4)(x+4)(x+4)(x+4) is not a factor of the polynomial.

Final Answer

The final answer is: x(x−8)(x+8)\boxed{x(x-8)(x+8)}

Introduction

In our previous article, we explored the completely factored form of the polynomial x3−64xx^3 - 64x. We examined each option and determined that the correct completely factored form is x(x−8)(x+8)x(x-8)(x+8). In this article, we will answer some frequently asked questions related to the completely factored form of x3−64xx^3 - 64x.

Q&A

Q: What is the completely factored form of x3−64xx^3 - 64x?

A: The completely factored form of x3−64xx^3 - 64x is x(x−8)(x+8)x(x-8)(x+8).

Q: Why is option A not the correct completely factored form?

A: Option A is not the correct completely factored form because the polynomial x3−64xx^3 - 64x does not have a repeated factor of (x−8)(x-8).

Q: Why is option B not the correct completely factored form?

A: Option B is not the correct completely factored form because the polynomial x3−64xx^3 - 64x does not have a factor of (x−4)(x-4), and the quadratic expression x2+4x+16x^2+4x+16 is not a factor of the polynomial.

Q: Why is option D not the correct completely factored form?

A: Option D is not the correct completely factored form because the polynomial x3−64xx^3 - 64x does not have a factor of (x−4)(x-4), and the quadratic expression (x+4)(x+4)(x+4)(x+4) is not a factor of the polynomial.

Q: How do I factor a polynomial like x3−64xx^3 - 64x?

A: To factor a polynomial like x3−64xx^3 - 64x, you can start by looking for a greatest common factor (GCF) of the terms. In this case, the GCF is xx. Then, you can try to factor the remaining expression, which is x2−64x^2 - 64. You can recognize that 6464 is a perfect square, which is 828^2. Therefore, you can rewrite the expression as x(x−8)(x+8)x(x-8)(x+8).

Q: What is the difference between a completely factored form and a partially factored form?

A: A completely factored form is a product of binomials, where each binomial is a factor of the polynomial. A partially factored form is a product of binomials, where not all of the binomials are factors of the polynomial.

Q: How do I determine if a polynomial is completely factored?

A: To determine if a polynomial is completely factored, you can try to factor it further. If you cannot factor it further, then it is completely factored.

Conclusion

In conclusion, the completely factored form of the polynomial x3−64xx^3 - 64x is x(x−8)(x+8)x(x-8)(x+8). We answered some frequently asked questions related to the completely factored form of x3−64xx^3 - 64x, and we provided some tips on how to factor polynomials.

Frequently Asked Questions

  • What is the completely factored form of x3−64xx^3 - 64x? The completely factored form of x3−64xx^3 - 64x is x(x−8)(x+8)x(x-8)(x+8).
  • Why is option A not the correct completely factored form? Option A is not the correct completely factored form because the polynomial x3−64xx^3 - 64x does not have a repeated factor of (x−8)(x-8).
  • Why is option B not the correct completely factored form? Option B is not the correct completely factored form because the polynomial x3−64xx^3 - 64x does not have a factor of (x−4)(x-4), and the quadratic expression x2+4x+16x^2+4x+16 is not a factor of the polynomial.
  • Why is option D not the correct completely factored form? Option D is not the correct completely factored form because the polynomial x3−64xx^3 - 64x does not have a factor of (x−4)(x-4), and the quadratic expression (x+4)(x+4)(x+4)(x+4) is not a factor of the polynomial.
  • How do I factor a polynomial like x3−64xx^3 - 64x? To factor a polynomial like x3−64xx^3 - 64x, you can start by looking for a greatest common factor (GCF) of the terms. In this case, the GCF is xx. Then, you can try to factor the remaining expression, which is x2−64x^2 - 64. You can recognize that 6464 is a perfect square, which is 828^2. Therefore, you can rewrite the expression as x(x−8)(x+8)x(x-8)(x+8).
  • What is the difference between a completely factored form and a partially factored form? A completely factored form is a product of binomials, where each binomial is a factor of the polynomial. A partially factored form is a product of binomials, where not all of the binomials are factors of the polynomial.
  • How do I determine if a polynomial is completely factored? To determine if a polynomial is completely factored, you can try to factor it further. If you cannot factor it further, then it is completely factored.

Final Answer

The final answer is: x(x−8)(x+8)\boxed{x(x-8)(x+8)}