What Is The Greatest Common Factor Of The Terms Of The Polynomial Below?${ 20x^4 - 10x^3 + 15x^2 }$A. ${ 5x^3 }$B. ${ 10x^3 }$C. ${ 10x^2 }$D. ${ 5x^2 }$

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What is the Greatest Common Factor of the Terms of the Polynomial?

Understanding the Concept of Greatest Common Factor (GCF)

The greatest common factor (GCF) of a set of numbers or terms is the largest expression that divides each of the numbers or terms without leaving a remainder. In the context of polynomials, the GCF is the largest expression that divides each term of the polynomial without leaving a remainder.

Identifying the Terms of the Polynomial

The given polynomial is:

20x4−10x3+15x2{ 20x^4 - 10x^3 + 15x^2 }

The terms of the polynomial are:

  • 20x420x^4
  • −10x3-10x^3
  • 15x215x^2

Finding the Greatest Common Factor

To find the GCF of the terms, we need to identify the common factors among them. The common factors are the factors that appear in each term.

  • The first term, 20x420x^4, has factors of 2020, x4x^4, and xx.
  • The second term, −10x3-10x^3, has factors of −10-10, x3x^3, and xx.
  • The third term, 15x215x^2, has factors of 1515, x2x^2, and xx.

Identifying the Common Factors

The common factors among the terms are:

  • xx (since it appears in each term)
  • 55 (since it is a factor of 2020, −10-10, and 1515)

Determining the Greatest Common Factor

Since xx and 55 are the common factors, we need to determine which one is the greatest. The greatest common factor is the product of the common factors, which is:

5x{ 5x }

However, we need to consider the exponents of the variables. The term with the highest exponent of xx is 20x420x^4. Therefore, the greatest common factor must include x4x^4.

The Greatest Common Factor

The greatest common factor of the terms of the polynomial is:

5x3{ 5x^3 }

Conclusion

The greatest common factor of the terms of the polynomial 20x4−10x3+15x220x^4 - 10x^3 + 15x^2 is 5x35x^3. This is the largest expression that divides each term of the polynomial without leaving a remainder.

Answer

The correct answer is:

A. 5x35x^3

Why is this the correct answer?

This is the correct answer because 5x35x^3 is the largest expression that divides each term of the polynomial without leaving a remainder. It is the product of the common factors, 55 and x3x^3, and it includes the highest exponent of xx.

What is the significance of the greatest common factor?

The greatest common factor is significant because it allows us to simplify polynomials by factoring out the common factors. This can make it easier to solve equations and perform other operations with polynomials.

How to find the greatest common factor

To find the greatest common factor, we need to identify the common factors among the terms of the polynomial. We can do this by listing the factors of each term and identifying the common factors. We then need to determine which one is the greatest by considering the exponents of the variables.

Tips and Tricks

  • Make sure to consider the exponents of the variables when determining the greatest common factor.
  • Use the distributive property to factor out the common factors.
  • Simplify the polynomial by factoring out the greatest common factor.

Real-World Applications

The greatest common factor has many real-world applications, including:

  • Simplifying polynomials in algebra
  • Factoring out common factors in geometry
  • Solving equations in calculus

Conclusion

In conclusion, the greatest common factor of the terms of the polynomial 20x4−10x3+15x220x^4 - 10x^3 + 15x^2 is 5x35x^3. This is the largest expression that divides each term of the polynomial without leaving a remainder. The greatest common factor is significant because it allows us to simplify polynomials by factoring out the common factors.
Q&A: Greatest Common Factor (GCF)

Frequently Asked Questions

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

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

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

A: To find the GCF of a polynomial, you need to identify the common factors among the terms of the polynomial. You can do this by listing the factors of each term and identifying the common factors. You then need to determine which one is the greatest by considering the exponents of the variables.

Q: What are the common factors of a polynomial?

A: The common factors of a polynomial are the factors that appear in each term of the polynomial. These can include variables, constants, and combinations of both.

Q: How do I determine the GCF of a polynomial with multiple terms?

A: To determine the GCF of a polynomial with multiple terms, you need to identify the common factors among all the terms. You can do this by listing the factors of each term and identifying the common factors. You then need to determine which one is the greatest by considering the exponents of the variables.

Q: What is the significance of the GCF in algebra?

A: The GCF is significant in algebra because it allows us to simplify polynomials by factoring out the common factors. This can make it easier to solve equations and perform other operations with polynomials.

Q: How do I use the GCF to simplify a polynomial?

A: To use the GCF to simplify a polynomial, you need to factor out the GCF from each term of the polynomial. This will result in a simplified polynomial with fewer terms.

Q: What are some common mistakes to avoid when finding the GCF of a polynomial?

A: Some common mistakes to avoid when finding the GCF of a polynomial include:

  • Not considering the exponents of the variables
  • Not identifying all the common factors
  • Not determining the greatest common factor

Q: How do I check my answer for the GCF of a polynomial?

A: To check your answer for the GCF of a polynomial, you need to make sure that the GCF divides each term of the polynomial without leaving a remainder. You can do this by dividing each term of the polynomial by the GCF and checking if the result is a whole number.

Q: What are some real-world applications of the GCF in algebra?

A: Some real-world applications of the GCF in algebra include:

  • Simplifying polynomials in algebra
  • Factoring out common factors in geometry
  • Solving equations in calculus

Q: How do I apply the GCF to solve equations in algebra?

A: To apply the GCF to solve equations in algebra, you need to factor out the GCF from each term of the equation. This will result in a simplified equation that is easier to solve.

Q: What are some tips and tricks for finding the GCF of a polynomial?

A: Some tips and tricks for finding the GCF of a polynomial include:

  • Make sure to consider the exponents of the variables
  • Use the distributive property to factor out the common factors
  • Simplify the polynomial by factoring out the GCF

Conclusion

In conclusion, the greatest common factor (GCF) is an important concept in algebra that allows us to simplify polynomials by factoring out the common factors. By understanding how to find the GCF and applying it to solve equations, you can become more proficient in algebra and solve problems more efficiently.

Additional Resources

  • For more information on the GCF, visit the Khan Academy website.
  • For practice problems and exercises, visit the Mathway website.
  • For a comprehensive guide to algebra, visit the Algebra.com website.

Final Thoughts

The GCF is a powerful tool in algebra that can help you simplify polynomials and solve equations more efficiently. By understanding how to find the GCF and applying it to solve problems, you can become more proficient in algebra and achieve your goals.