Factor Out The GCF Of The Three Terms, Then Complete The Factorization Of 6 X 3 + 6 X 2 − 120 X 6x^3 + 6x^2 - 120x 6 X 3 + 6 X 2 − 120 X .A. 6 X ( X 2 + X − 20 6x(x^2 + X - 20 6 X ( X 2 + X − 20 ] B. 6 X ( X − 4 ) ( X + 5 6x(x-4)(x+5 6 X ( X − 4 ) ( X + 5 ] C. 6 X 2 ( X + 1 ) − 120 X 6x^2(x+1) - 120x 6 X 2 ( X + 1 ) − 120 X D. 6 X ( X + 4 ) ( X − 5 6x(x+4)(x-5 6 X ( X + 4 ) ( X − 5 ]

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Introduction

Factoring polynomials is an essential skill in algebra, and it requires a deep understanding of the properties of polynomials. In this article, we will focus on factoring the polynomial 6x3+6x2−120x6x^3 + 6x^2 - 120x by first identifying the greatest common factor (GCF) of the three terms and then completing the factorization.

Identifying the GCF

To factor out the GCF, we need to identify the common factors among the three terms. The GCF is the largest factor that divides all three terms without leaving a remainder. In this case, we can see that all three terms have a common factor of 6x6x. Therefore, we can factor out 6x6x from each term.

Factoring out the GCF

To factor out the GCF, we need to divide each term by the GCF. In this case, we will divide each term by 6x6x.

6x36x=x2\frac{6x^3}{6x} = x^2

6x26x=x\frac{6x^2}{6x} = x

−120x6x=−20\frac{-120x}{6x} = -20

Completing the Factorization

Now that we have factored out the GCF, we can complete the factorization by multiplying the GCF with the remaining factors.

6x(x2+x−20)6x(x^2 + x - 20)

Checking the Answer Choices

Now that we have completed the factorization, we can check the answer choices to see which one matches our result.

A. 6x(x2+x−20)6x(x^2 + x - 20)

B. 6x(x−4)(x+5)6x(x-4)(x+5)

C. 6x2(x+1)−120x6x^2(x+1) - 120x

D. 6x(x+4)(x−5)6x(x+4)(x-5)

Conclusion

Based on our calculations, we can see that the correct answer is A. 6x(x2+x−20)6x(x^2 + x - 20). This is because our factorization matches the answer choice exactly.

Tips and Tricks

When factoring polynomials, it's essential to identify the GCF and factor it out first. This will make it easier to complete the factorization and ensure that you get the correct answer.

Common Mistakes

One common mistake when factoring polynomials is to forget to factor out the GCF. This can lead to incorrect factorization and incorrect answers. To avoid this mistake, make sure to identify the GCF and factor it out first.

Real-World Applications

Factoring polynomials has many real-world applications, including:

  • Science and Engineering: Factoring polynomials is used to solve equations and model real-world phenomena.
  • Computer Science: Factoring polynomials is used in computer algorithms and data analysis.
  • Economics: Factoring polynomials is used to model economic systems and make predictions.

Final Thoughts

Factoring polynomials is an essential skill in algebra, and it requires a deep understanding of the properties of polynomials. By identifying the GCF and factoring it out, we can complete the factorization and get the correct answer. Remember to always check the answer choices and make sure to factor out the GCF first.

Factorization Techniques

There are several techniques used to factor polynomials, including:

  • GCF Factoring: This involves factoring out the greatest common factor (GCF) of the terms.
  • Difference of Squares: This involves factoring the difference of two squares.
  • Perfect Square Trinomials: This involves factoring perfect square trinomials.

Real-World Examples

Factoring polynomials has many real-world applications, including:

  • Science and Engineering: Factoring polynomials is used to solve equations and model real-world phenomena.
  • Computer Science: Factoring polynomials is used in computer algorithms and data analysis.
  • Economics: Factoring polynomials is used to model economic systems and make predictions.

Conclusion

Factoring polynomials is an essential skill in algebra, and it requires a deep understanding of the properties of polynomials. By identifying the GCF and factoring it out, we can complete the factorization and get the correct answer. Remember to always check the answer choices and make sure to factor out the GCF first.

Final Tips

When factoring polynomials, make sure to:

  • Identify the GCF: Factor out the greatest common factor (GCF) of the terms.
  • Check the Answer Choices: Make sure to check the answer choices and see which one matches your result.
  • Use Factoring Techniques: Use factoring techniques such as GCF factoring, difference of squares, and perfect square trinomials to factor polynomials.

References

  • Algebra: A Comprehensive Introduction by Michael Artin
  • Calculus: Early Transcendentals by James Stewart
  • Linear Algebra and Its Applications by Gilbert Strang

Introduction

Factoring polynomials is an essential skill in algebra, and it requires a deep understanding of the properties of polynomials. In this article, we will answer some common questions about factoring polynomials and provide tips and tricks to help you master this skill.

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

A: The greatest common factor (GCF) of a polynomial is the largest factor that divides all the terms of the polynomial without leaving a remainder.

Q: How do I identify the GCF of a polynomial?

A: To identify the GCF of a polynomial, you need to look for the common factors among the terms. You can do this by listing the factors of each term and finding the greatest common factor.

Q: What is the difference of squares?

A: The difference of squares is a factoring technique used to factor the difference of two squares. It is represented as:

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)

Q: How do I factor a difference of squares?

A: To factor a difference of squares, you need to identify the two squares and then factor them using the difference of squares formula.

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a trinomial that can be factored into the square of a binomial. It is represented as:

a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a + b)^2

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, you need to identify the two binomials and then factor them using the perfect square trinomial formula.

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

A: Some common mistakes to avoid when factoring polynomials include:

  • Forgetting to factor out the GCF: Make sure to factor out the greatest common factor (GCF) of the terms.
  • Not checking the answer choices: Make sure to check the answer choices and see which one matches your result.
  • Using the wrong factoring technique: Make sure to use the correct factoring technique for the type of polynomial you are factoring.

Q: How do I practice factoring polynomials?

A: To practice factoring polynomials, you can try the following:

  • Practice factoring simple polynomials: Start with simple polynomials and practice factoring them using the GCF, difference of squares, and perfect square trinomial techniques.
  • Use online resources: There are many online resources available that can help you practice factoring polynomials, including worksheets, quizzes, and games.
  • Work with a tutor or teacher: Working with a tutor or teacher can help you get personalized feedback and guidance on factoring polynomials.

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

A: Factoring polynomials has many real-world applications, including:

  • Science and Engineering: Factoring polynomials is used to solve equations and model real-world phenomena.
  • Computer Science: Factoring polynomials is used in computer algorithms and data analysis.
  • Economics: Factoring polynomials is used to model economic systems and make predictions.

Q: How do I know if I am ready to move on to more advanced topics in algebra?

A: To know if you are ready to move on to more advanced topics in algebra, you need to demonstrate a strong understanding of the basics of algebra, including factoring polynomials. You should be able to:

  • Factor simple polynomials: You should be able to factor simple polynomials using the GCF, difference of squares, and perfect square trinomial techniques.
  • Solve equations: You should be able to solve equations using algebraic techniques.
  • Model real-world phenomena: You should be able to model real-world phenomena using algebraic techniques.

Conclusion

Factoring polynomials is an essential skill in algebra, and it requires a deep understanding of the properties of polynomials. By practicing factoring polynomials and using the correct factoring techniques, you can master this skill and move on to more advanced topics in algebra.

Final Tips

When factoring polynomials, make sure to:

  • Identify the GCF: Factor out the greatest common factor (GCF) of the terms.
  • Check the answer choices: Make sure to check the answer choices and see which one matches your result.
  • Use factoring techniques: Use factoring techniques such as GCF factoring, difference of squares, and perfect square trinomials to factor polynomials.

References

  • Algebra: A Comprehensive Introduction by Michael Artin
  • Calculus: Early Transcendentals by James Stewart
  • Linear Algebra and Its Applications by Gilbert Strang