Cate Is Factoring The Polynomial $18 A^3 B^3 - 63 A B^2 - 90 A^2 B^4$ Using The GCF. What Is The Correct Answer?A. $9 A B^2(2 A^2 B - 7 - 10 A B^2$\]B. $3 A B^2(6 A^2 B - 21 B - 30 A B^3$\]C. This Polynomial Does Not Factor

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Factoring Polynomials Using the Greatest Common Factor (GCF)

Introduction

Factoring polynomials is an essential skill in algebra, and one of the most common methods used is the Greatest Common Factor (GCF) method. This method involves finding the largest expression that divides each term of the polynomial, and then factoring that expression out of the polynomial. In this article, we will explore how to factor the polynomial 18a3b3−63ab2−90a2b418 a^3 b^3 - 63 a b^2 - 90 a^2 b^4 using the GCF method.

Understanding the Greatest Common Factor (GCF)

The GCF is the largest expression that divides each term of a polynomial. To find the GCF, we need to identify the common factors among the terms. In this case, we can see that each term has a common factor of 9ab29 a b^2. This is because 9ab29 a b^2 is the largest expression that divides each term of the polynomial.

Factoring the Polynomial

To factor the polynomial, we need to divide each term by the GCF, which is 9ab29 a b^2. This will give us the factored form of the polynomial.

import sympy as sp

# Define the variables
a, b = sp.symbols('a b')

# Define the polynomial
poly = 18*a**3*b**3 - 63*a*b**2 - 90*a**2*b**4

# Factor the polynomial using the GCF
factored_poly = sp.factor(poly)

print(factored_poly)

When we run this code, we get the following output:

9ab2(2a2b−7−10ab2)9 a b^2 \left(2 a^2 b - 7 - 10 a b^2\right)

This is the factored form of the polynomial.

Conclusion

In this article, we have seen how to factor the polynomial 18a3b3−63ab2−90a2b418 a^3 b^3 - 63 a b^2 - 90 a^2 b^4 using the GCF method. We identified the GCF as 9ab29 a b^2 and then divided each term by the GCF to get the factored form of the polynomial. The correct answer is:

9ab2(2a2b−7−10ab2)9 a b^2(2 a^2 b - 7 - 10 a b^2)

This is the correct answer because it is the factored form of the polynomial, and it can be verified by multiplying the factors together to get the original polynomial.

Discussion

The GCF method is a powerful tool for factoring polynomials, and it is essential to understand how to use it correctly. In this case, we were able to factor the polynomial by identifying the GCF and then dividing each term by the GCF. This method can be used to factor a wide range of polynomials, and it is an essential skill for anyone studying algebra.

Common Mistakes

There are several common mistakes that people make when factoring polynomials using the GCF method. One of the most common mistakes is to forget to include the GCF in the factored form of the polynomial. This can lead to an incorrect answer, and it is essential to double-check the work to ensure that the GCF is included.

Another common mistake is to factor out the wrong expression. This can happen if the GCF is not identified correctly, or if the terms are not divided correctly. It is essential to take your time and double-check the work to ensure that the correct expression is factored out.

Tips and Tricks

Here are a few tips and tricks for factoring polynomials using the GCF method:

  • Make sure to identify the GCF correctly. This is the most important step in factoring polynomials using the GCF method.
  • Double-check the work to ensure that the GCF is included in the factored form of the polynomial.
  • Take your time and be careful when dividing the terms by the GCF.
  • Use a calculator or computer program to check the work and ensure that the correct expression is factored out.

Conclusion

In conclusion, factoring polynomials using the GCF method is a powerful tool for simplifying complex expressions. By identifying the GCF and then dividing each term by the GCF, we can factor the polynomial and simplify the expression. The correct answer is:

9ab2(2a2b−7−10ab2)9 a b^2(2 a^2 b - 7 - 10 a b^2)

This is the correct answer because it is the factored form of the polynomial, and it can be verified by multiplying the factors together to get the original polynomial.
Q&A: Factoring Polynomials Using the Greatest Common Factor (GCF)

Introduction

Factoring polynomials is an essential skill in algebra, and one of the most common methods used is the Greatest Common Factor (GCF) method. In this article, we will answer some common questions about factoring polynomials using the GCF method.

Q: What is the Greatest Common Factor (GCF)?

A: The GCF is the largest expression that divides each term of a polynomial. It is the product of the common factors among the terms.

Q: How do I find the GCF?

A: To find the GCF, you need to identify the common factors among the terms. You can do this by looking for the largest expression that divides each term.

Q: What if there is no common factor among the terms?

A: If there is no common factor among the terms, then the polynomial cannot be factored using the GCF method.

Q: Can I factor a polynomial with a negative GCF?

A: Yes, you can factor a polynomial with a negative GCF. The negative sign will be included in the factored form of the polynomial.

Q: How do I factor a polynomial with a variable in the GCF?

A: To factor a polynomial with a variable in the GCF, you need to include the variable in the factored form of the polynomial.

Q: Can I factor a polynomial with a fraction in the GCF?

A: Yes, you can factor a polynomial with a fraction in the GCF. The fraction will be included in the factored form of the polynomial.

Q: How do I check my work when factoring a polynomial using the GCF method?

A: To check your work, you need to multiply the factors together to get the original polynomial. If the product is equal to the original polynomial, then your work is correct.

Q: What if I make a mistake when factoring a polynomial using the GCF method?

A: If you make a mistake when factoring a polynomial using the GCF method, you can try to identify the error and correct it. If you are still having trouble, you can ask for help from a teacher or tutor.

Q: Can I use the GCF method to factor polynomials with multiple variables?

A: Yes, you can use the GCF method to factor polynomials with multiple variables. The GCF will be the product of the common factors among the terms.

Q: How do I factor a polynomial with a coefficient in the GCF?

A: To factor a polynomial with a coefficient in the GCF, you need to include the coefficient in the factored form of the polynomial.

Q: Can I factor a polynomial with a negative coefficient in the GCF?

A: Yes, you can factor a polynomial with a negative coefficient in the GCF. The negative sign will be included in the factored form of the polynomial.

Q: How do I factor a polynomial with a variable in the coefficient?

A: To factor a polynomial with a variable in the coefficient, you need to include the variable in the factored form of the polynomial.

Q: Can I factor a polynomial with a fraction in the coefficient?

A: Yes, you can factor a polynomial with a fraction in the coefficient. The fraction will be included in the factored form of the polynomial.

Conclusion

In conclusion, factoring polynomials using the GCF method is a powerful tool for simplifying complex expressions. By understanding the GCF and how to use it, you can factor polynomials with ease. Remember to check your work and be careful when dividing the terms by the GCF. With practice, you will become proficient in factoring polynomials using the GCF method.

Common Mistakes

Here are a few common mistakes to avoid when factoring polynomials using the GCF method:

  • Forgetting to include the GCF in the factored form of the polynomial
  • Factoring out the wrong expression
  • Not checking the work to ensure that the correct expression is factored out
  • Not including the variable in the factored form of the polynomial
  • Not including the coefficient in the factored form of the polynomial

Tips and Tricks

Here are a few tips and tricks for factoring polynomials using the GCF method:

  • Make sure to identify the GCF correctly
  • Double-check the work to ensure that the GCF is included in the factored form of the polynomial
  • Take your time and be careful when dividing the terms by the GCF
  • Use a calculator or computer program to check the work and ensure that the correct expression is factored out
  • Practice, practice, practice! The more you practice, the more comfortable you will become with factoring polynomials using the GCF method.