What Is The Factored Form Of $6n^4 - 24n^3 + 18n?$A. $6n(n^4 + 4n^3 + 3n)$ B. \$6n(n^4 - 4n^3 + 3n)$[/tex\] C. $6n(n^3 - 4n^2 + 3)$ D. $6n(n^3 + 4n^2 + 3)$

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Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will explore the factored form of the polynomial $6n^4 - 24n^3 + 18n$ and examine the different options provided.

Understanding the Polynomial

The given polynomial is $6n^4 - 24n^3 + 18n$. To factor this polynomial, we need to identify the greatest common factor (GCF) of the terms. The GCF of the terms is 6n, which can be factored out.

Factoring Out the GCF

To factor out the GCF, we divide each term by 6n.

6n46n=n3\frac{6n^4}{6n} = n^3

−24n36n=−4n2\frac{-24n^3}{6n} = -4n^2

18n6n=3\frac{18n}{6n} = 3

Now, we can rewrite the polynomial as:

6n4−24n3+18n=6n(n3−4n2+3)6n^4 - 24n^3 + 18n = 6n(n^3 - 4n^2 + 3)

Analyzing the Options

Now that we have factored the polynomial, let's analyze the options provided.

  • Option A: $6n(n^4 + 4n^3 + 3n)$ This option is incorrect because the polynomial inside the parentheses is not the correct factorization of the original polynomial.
  • Option B: $6n(n^4 - 4n^3 + 3n)$ This option is incorrect because the polynomial inside the parentheses is not the correct factorization of the original polynomial.
  • Option C: $6n(n^3 - 4n^2 + 3)$ This option is correct because it matches the factored form of the polynomial that we obtained earlier.
  • Option D: $6n(n^3 + 4n^2 + 3)$ This option is incorrect because the polynomial inside the parentheses is not the correct factorization of the original polynomial.

Conclusion

In conclusion, the factored form of the polynomial $6n^4 - 24n^3 + 18n$ is $6n(n^3 - 4n^2 + 3)$. This is the correct option among the ones provided.

Tips and Tricks

When factoring polynomials, it's essential to identify the greatest common factor (GCF) of the terms and factor it out. This will help you simplify the polynomial and make it easier to work with.

Common Mistakes to Avoid

When factoring polynomials, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not identifying the GCF: Failing to identify the GCF of the terms can lead to incorrect factorization.
  • Not factoring out the GCF: Failing to factor out the GCF can lead to incorrect factorization.
  • Not checking the options: Failing to check the options can lead to selecting the incorrect answer.

Real-World Applications

Factoring polynomials has numerous real-world applications. Here are a few examples:

  • Solving equations: Factoring polynomials is essential for solving equations, especially quadratic equations.
  • Graphing functions: Factoring polynomials is essential for graphing functions, especially polynomial functions.
  • Optimization problems: Factoring polynomials is essential for solving optimization problems, especially those involving quadratic functions.

Final Thoughts

Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will answer some frequently asked questions about factoring polynomials.

Q: What is factoring a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials. This is done by identifying the greatest common factor (GCF) of the terms and factoring it out.

Q: Why is factoring a polynomial important?

A: Factoring a polynomial is important because it helps to simplify the polynomial and make it easier to work with. It is also essential for solving equations, graphing functions, and solving optimization problems.

Q: How do I factor a polynomial?

A: To factor a polynomial, you need to identify the greatest common factor (GCF) of the terms and factor it out. This involves dividing each term by the GCF and rewriting the polynomial as a product of simpler polynomials.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides each term of the polynomial. It is essential to identify the GCF to factor the polynomial correctly.

Q: How do I identify the GCF?

A: To identify the GCF, you need to look for the largest factor that divides each term of the polynomial. You can do this by listing the factors of each term and finding the largest factor that is common to all terms.

Q: What are some common mistakes to avoid when factoring a polynomial?

A: Some common mistakes to avoid when factoring a polynomial include:

  • Not identifying the GCF
  • Not factoring out the GCF
  • Not checking the options
  • Not simplifying the polynomial

Q: How do I check my work when factoring a polynomial?

A: To check your work when factoring a polynomial, you need to:

  • Verify that the GCF is correct
  • Verify that the polynomial is factored correctly
  • Simplify the polynomial to ensure that it is in its simplest form

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

A: Some real-world applications of factoring polynomials include:

  • Solving equations
  • Graphing functions
  • Solving optimization problems
  • Analyzing data

Q: Can I use technology to help me factor polynomials?

A: Yes, you can use technology to help you factor polynomials. There are many online tools and software programs available that can help you factor polynomials quickly and accurately.

Q: How do I choose the correct factoring method?

A: To choose the correct factoring method, you need to consider the type of polynomial you are working with and the level of difficulty. Some common factoring methods include:

  • Factoring out the GCF
  • Factoring by grouping
  • Factoring quadratics
  • Factoring polynomials with rational exponents

Conclusion

In conclusion, factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. By identifying the greatest common factor (GCF) of the terms and factoring it out, we can simplify the polynomial and make it easier to work with. With practice and patience, you can master the art of factoring polynomials and apply it to real-world problems.

Tips and Tricks

Here are some tips and tricks to help you factor polynomials:

  • Practice, practice, practice: The more you practice factoring polynomials, the more comfortable you will become with the process.
  • Use technology: There are many online tools and software programs available that can help you factor polynomials quickly and accurately.
  • Simplify the polynomial: Make sure to simplify the polynomial to ensure that it is in its simplest form.
  • Check your work: Verify that the GCF is correct and that the polynomial is factored correctly.