What Is The Complete Factorization Of X 3 + 64 X^3 + 64 X 3 + 64 ?A. (x+4)\left(x^2+4x+16\right ] B. (x+4)\left(x^2-4x+16\right ] C. (x-4)\left(x^2-4x+16\right ] D. (x-4)\left(x^2+4x+16\right ]

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Introduction

In this article, we will delve into the world of algebra and explore the complete factorization of the polynomial $x^3 + 64$. Factorization is a fundamental concept in mathematics, and it plays a crucial role in solving equations, finding roots, and simplifying expressions. The complete factorization of a polynomial is the process of expressing it as a product of simpler polynomials, known as factors. In this case, we will examine the factorization of $x^3 + 64$ and explore the different options provided.

Understanding the Polynomial

Before we proceed with the factorization, let's take a closer look at the polynomial $x^3 + 64$. This is a cubic polynomial, which means it has a degree of 3. The polynomial can be written as $x^3 + 64 = (x^3) + 64$. We can see that the polynomial has a constant term of 64, which is a perfect cube of 4.

Identifying the Factor

To factorize the polynomial, we need to identify a factor that, when multiplied by another factor, will result in the original polynomial. In this case, we can start by looking for a factor that, when multiplied by itself, will result in $x^3 + 64$. We can see that $x+4$ is a factor of the polynomial, as it satisfies the condition.

Factoring the Polynomial

Now that we have identified the factor $x+4$, we can proceed with the factorization. We can write the polynomial as $(x+4)(x^2+4x+16)$. This is the complete factorization of the polynomial $x^3 + 64$.

Exploring the Options

Let's take a closer look at the options provided:

• A. $(x+4)\left(x^2+4x+16\right)$
• B. $(x+4)\left(x^2-4x+16\right)$
• C. $(x-4)\left(x^2-4x+16\right)$
• D. $(x-4)\left(x^2+4x+16\right)$

We can see that option A is the correct factorization of the polynomial $x^3 + 64$. The other options are incorrect, as they do not result in the original polynomial.

Conclusion

In this article, we have explored the complete factorization of the polynomial $x^3 + 64$. We have identified the factor $x+4$ and proceeded with the factorization. We have also examined the options provided and concluded that option A is the correct factorization of the polynomial.

Final Answer

The complete factorization of $x^3 + 64$ is:

$(x+4)\left(x^2+4x+16\right)$

This is the correct answer, and it can be verified by multiplying the factors together.

Frequently Asked Questions

• Q: What is the complete factorization of $x^3 + 64$? A: The complete factorization of $x^3 + 64$ is $(x+4)\left(x^2+4x+16\right)$.
• Q: How do I factorize a polynomial? A: To factorize a polynomial, you need to identify a factor that, when multiplied by another factor, will result in the original polynomial.
• Q: What is the degree of the polynomial $x^3 + 64$? A: The degree of the polynomial $x^3 + 64$ is 3.

References

• [1] Algebra, by Michael Artin
• [2] Calculus, by Michael Spivak
• [3] Mathematics, by Richard Courant

Related Articles

• [1] Factorization of Polynomials
• [2] Roots of Polynomials
• [3] Simplifying Expressions

Tags

• Factorization
• Polynomials
• Algebra
• Mathematics
• Roots
• Simplifying Expressions
  

Introduction

In our previous article, we explored the complete factorization of the polynomial $x^3 + 64$. Factorization is a fundamental concept in mathematics, and it plays a crucial role in solving equations, finding roots, and simplifying expressions. In this article, we will answer some frequently asked questions about factorization of polynomials.

Q&A

Q: What is factorization?

A: Factorization is the process of expressing a polynomial as a product of simpler polynomials, known as factors.

Q: Why is factorization important?

A: Factorization is important because it helps us to solve equations, find roots, and simplify expressions. It is a fundamental concept in mathematics and is used in many areas of mathematics, including algebra, geometry, and calculus.

Q: How do I factorize a polynomial?

A: To factorize a polynomial, you need to identify a factor that, when multiplied by another factor, will result in the original polynomial. You can use various techniques, such as grouping, synthetic division, and the rational root theorem, to factorize a polynomial.

Q: What is the difference between factoring and simplifying?

A: Factoring and simplifying are two different processes. Factoring involves expressing a polynomial as a product of simpler polynomials, while simplifying involves reducing a polynomial to its simplest form.

Q: Can you give an example of factoring a polynomial?

A: Yes, let's consider the polynomial $x^2 + 5x + 6$. We can factor this polynomial as $(x+2)(x+3)$.

Q: How do I know if a polynomial can be factored?

A: You can use various techniques, such as the rational root theorem, to determine if a polynomial can be factored. If the polynomial has a rational root, it can be factored.

Q: What is the rational root theorem?

A: The rational root theorem states that if a polynomial has a rational root, then that root must be a factor of the constant term divided by a factor of the leading coefficient.

Q: Can you give an example of using the rational root theorem?

A: Yes, let's consider the polynomial $x^3 + 2x^2 + 3x + 1$. We can use the rational root theorem to determine if this polynomial has a rational root. The constant term is 1, and the leading coefficient is 1. Therefore, the possible rational roots are $\pm 1$.

Q: How do I use synthetic division to factor a polynomial?

A: Synthetic division is a technique used to factor a polynomial by dividing it by a linear factor. You can use synthetic division to factor a polynomial by following these steps:

1. Write the polynomial in the form $ax^2 + bx + c$.
2. Write the linear factor in the form $x - r$.
3. Perform synthetic division by dividing the polynomial by the linear factor.
4. The result of the synthetic division will be the factored form of the polynomial.

Q: Can you give an example of using synthetic division?

A: Yes, let's consider the polynomial $x^3 + 2x^2 + 3x + 1$. We can use synthetic division to factor this polynomial by dividing it by the linear factor $x - 1$.

Conclusion

In this article, we have answered some frequently asked questions about factorization of polynomials. We have discussed the importance of factorization, the different techniques used to factorize a polynomial, and the rational root theorem. We have also provided examples of factoring a polynomial using synthetic division.

Final Answer

Factorization is a fundamental concept in mathematics, and it plays a crucial role in solving equations, finding roots, and simplifying expressions. By understanding the different techniques used to factorize a polynomial, you can solve equations and simplify expressions more efficiently.

Frequently Asked Questions

• Q: What is factorization? A: Factorization is the process of expressing a polynomial as a product of simpler polynomials, known as factors.
• Q: Why is factorization important? A: Factorization is important because it helps us to solve equations, find roots, and simplify expressions.
• Q: How do I factorize a polynomial? A: To factorize a polynomial, you need to identify a factor that, when multiplied by another factor, will result in the original polynomial.

References

• [1] Algebra, by Michael Artin
• [2] Calculus, by Michael Spivak
• [3] Mathematics, by Richard Courant

Related Articles

• [1] Factorization of Polynomials
• [2] Roots of Polynomials
• [3] Simplifying Expressions

Tags

• Factorization
• Polynomials
• Algebra
• Mathematics
• Roots
• Simplifying Expressions