What Is The Greatest Common Factor Of $12xy^5 + 60x^4y^2 - 24x^3y^3$?A. $6xy^2$B. \$12xy^2$[/tex\]C. $6xy^5$D. $3x^4y^5$

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Introduction

In algebra, the greatest common factor (GCF) of a polynomial expression is the largest expression that divides each term of the polynomial without leaving a remainder. Finding the GCF of a polynomial is an essential skill in mathematics, as it helps us simplify complex expressions and solve equations. In this article, we will explore the concept of GCF and apply it to a specific polynomial expression.

What is a Polynomial Expression?

A polynomial expression is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial expression can be raised to non-negative integer powers. For example, the expression $12xy^5 + 60x4y2 - 24x3y3$ is a polynomial expression, where $x$ and $y$ are variables.

How to Find the Greatest Common Factor of a Polynomial Expression

To find the GCF of a polynomial expression, we need to identify the common factors among the terms. A common factor is a factor that divides each term of the polynomial without leaving a remainder. We can find the GCF by factoring each term of the polynomial and then identifying the common factors.

Step 1: Factor Each Term of the Polynomial

To find the GCF, we need to factor each term of the polynomial. We can factor out the greatest common factor of each term.

  • The first term, $12xy^5$, can be factored as $12xy^5 = 12xy^4 \cdot y$.
  • The second term, $60x4y2$, can be factored as $60x4y2 = 60x^3 \cdot x \cdot y^2$.
  • The third term, $-24x3y3$, can be factored as $-24x3y3 = -24x^2 \cdot x \cdot y^2 \cdot y$.

Step 2: Identify the Common Factors

Now that we have factored each term of the polynomial, we can identify the common factors. The common factors are the factors that appear in each term.

  • The common factors of the first term, $12xy^4 \cdot y$, are $12xy^4$.
  • The common factors of the second term, $60x^3 \cdot x \cdot y^2$, are $60x^3 \cdot x$.
  • The common factors of the third term, $-24x^2 \cdot x \cdot y^2 \cdot y$, are $-24x^2 \cdot x \cdot y^2$.

Step 3: Find the Greatest Common Factor

Now that we have identified the common factors, we can find the greatest common factor. The greatest common factor is the largest expression that divides each term of the polynomial without leaving a remainder.

  • The greatest common factor of the first term, $12xy^4 \cdot y$, is $12xy^4$.
  • The greatest common factor of the second term, $60x^3 \cdot x \cdot y^2$, is $60x^3 \cdot x$.
  • The greatest common factor of the third term, $-24x^2 \cdot x \cdot y^2 \cdot y$, is $-24x^2 \cdot x \cdot y^2$.

Conclusion

In conclusion, the greatest common factor of the polynomial expression $12xy^5 + 60x4y2 - 24x3y3$ is $6xy^2$. This is because $6xy^2$ is the largest expression that divides each term of the polynomial without leaving a remainder.

Answer

The correct answer is:

  • A. $6xy^2$

Why is this the correct answer?

This is the correct answer because $6xy^2$ is the greatest common factor of the polynomial expression $12xy^5 + 60x4y2 - 24x3y3$. This means that $6xy^2$ is the largest expression that divides each term of the polynomial without leaving a remainder.

What are the other options?

The other options are:

  • B. $12xy^2$
  • C. $6xy^5$
  • D. $3x4y5$

Why are these options incorrect?

These options are incorrect because they are not the greatest common factor of the polynomial expression $12xy^5 + 60x4y2 - 24x3y3$. For example, $12xy^2$ is not the greatest common factor because it does not divide the third term, $-24x3y3$, without leaving a remainder.

Conclusion

Q: What is the greatest common factor of a polynomial expression?

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

Q: How do I find the greatest common factor of a polynomial expression?

A: To find the GCF of a polynomial expression, you need to identify the common factors among the terms. You can do this by factoring each term of the polynomial and then identifying the common factors.

Q: What are the steps to find the greatest common factor of a polynomial expression?

A: The steps to find the GCF of a polynomial expression are:

  1. Factor each term of the polynomial.
  2. Identify the common factors.
  3. Find the greatest common factor.

Q: What is the greatest common factor of the polynomial expression $12xy^5 + 60x4y2 - 24x3y3$?

A: The greatest common factor of the polynomial expression $12xy^5 + 60x4y2 - 24x3y3$ is $6xy^2$.

Q: Why is $6xy^2$ the greatest common factor of the polynomial expression $12xy^5 + 60x4y2 - 24x3y3$?

A: $6xy^2$ is the greatest common factor of the polynomial expression $12xy^5 + 60x4y2 - 24x3y3$ because it is the largest expression that divides each term of the polynomial without leaving a remainder.

Q: What are the other options for the greatest common factor of the polynomial expression $12xy^5 + 60x4y2 - 24x3y3$?

A: The other options for the GCF of the polynomial expression $12xy^5 + 60x4y2 - 24x3y3$ are:

  • B. $12xy^2$
  • C. $6xy^5$
  • D. $3x4y5$

Q: Why are these options incorrect?

A: These options are incorrect because they are not the greatest common factor of the polynomial expression $12xy^5 + 60x4y2 - 24x3y3$. For example, $12xy^2$ is not the greatest common factor because it does not divide the third term, $-24x3y3$, without leaving a remainder.

Q: How do I know if I have found the greatest common factor of a polynomial expression?

A: You know if you have found the GCF of a polynomial expression if it is the largest expression that divides each term of the polynomial without leaving a remainder.

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

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

  • Not factoring each term of the polynomial.
  • Not identifying the common factors.
  • Not finding the greatest common factor.

Q: How can I practice finding the greatest common factor of a polynomial expression?

A: You can practice finding the GCF of a polynomial expression by working through examples and exercises. You can also use online resources and tools to help you practice.

Conclusion

In conclusion, finding the greatest common factor of a polynomial expression is an essential skill in mathematics. By following the steps outlined in this article, you can find the GCF of a polynomial expression and solve problems with confidence.