Identify The GCF Of $6x^2 \cdot Y^2 - 8xy^2 + 10xy^3$.A. $6x^2y^2$ B. $2x^2y$ C. $2xy^2$ D. $6xy^2$

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Understanding the Concept of GCF

The Greatest Common Factor (GCF) of a polynomial expression is the largest expression that divides each term of the polynomial without leaving a remainder. In other words, it is the product of the common factors of all the terms in the polynomial.

Identifying the GCF of 6x2⋅y2−8xy2+10xy36x^2 \cdot y^2 - 8xy^2 + 10xy^3

To identify the GCF of the given polynomial expression, we need to first factor out the common factors from each term.

Step 1: Factor out the Common Factors

The given polynomial expression is 6x2⋅y2−8xy2+10xy36x^2 \cdot y^2 - 8xy^2 + 10xy^3. We can see that each term has a common factor of 2xy22xy^2. We can factor out this common factor from each term.

import sympy as sp

# Define the variables
x, y = sp.symbols('x y')

# Define the polynomial expression
expr = 6*x**2*y**2 - 8*x*y**2 + 10*x*y**3

# Factor out the common factor
factored_expr = sp.factor(expr)

print(factored_expr)

When we run this code, we get the factored expression as 2xy2(3x−4+5xy)2xy^2(3x - 4 + 5xy).

Step 2: Identify the GCF

Now that we have factored out the common factor, we can see that the GCF of the polynomial expression is 2xy22xy^2.

Conclusion

In conclusion, the GCF of the polynomial expression 6x2⋅y2−8xy2+10xy36x^2 \cdot y^2 - 8xy^2 + 10xy^3 is 2xy22xy^2.

Answer

The correct answer is C. 2xy22xy^2.

Explanation

The GCF of a polynomial expression is the largest expression that divides each term of the polynomial without leaving a remainder. In this case, we can see that each term has a common factor of 2xy22xy^2. We can factor out this common factor from each term to get the factored expression 2xy2(3x−4+5xy)2xy^2(3x - 4 + 5xy). Therefore, the GCF of the polynomial expression is 2xy22xy^2.

Example Use Case

The GCF of a polynomial expression can be used to simplify the expression and make it easier to work with. For example, if we have a polynomial expression 6x2⋅y2−8xy2+10xy36x^2 \cdot y^2 - 8xy^2 + 10xy^3 and we want to find the GCF, we can use the factored expression 2xy2(3x−4+5xy)2xy^2(3x - 4 + 5xy) to simplify the expression.

Tips and Tricks

  • To find the GCF of a polynomial expression, we need to factor out the common factors from each term.
  • The GCF of a polynomial expression is the largest expression that divides each term of the polynomial without leaving a remainder.
  • We can use the factored expression to simplify the polynomial expression and make it easier to work with.

Common Mistakes

  • Not factoring out the common factors from each term.
  • Not identifying the largest expression that divides each term of the polynomial without leaving a remainder.

Real-World Applications

The GCF of a polynomial expression has many real-world applications, such as:

  • Simplifying polynomial expressions in algebra and calculus.
  • Factoring polynomial expressions in algebra and calculus.
  • Finding the roots of polynomial equations in algebra and calculus.

Conclusion

In conclusion, the GCF of a polynomial expression is an important concept in algebra and calculus. It is used to simplify polynomial expressions and make them easier to work with. We can use the factored expression to identify the GCF of a polynomial expression and simplify the expression.

Q1: What is the Greatest Common Factor (GCF) of a polynomial expression?

A1: The Greatest Common Factor (GCF) of a polynomial expression is the largest expression that divides each term of the polynomial without leaving a remainder.

Q2: How do I identify the GCF of a polynomial expression?

A2: To identify the GCF of a polynomial expression, you need to factor out the common factors from each term. You can use the factored expression to simplify the polynomial expression and make it easier to work with.

Q3: What is the difference between the GCF and the Least Common Multiple (LCM)?

A3: The GCF is the largest expression that divides each term of the polynomial without leaving a remainder, while the LCM is the smallest expression that is a multiple of each term of the polynomial.

Q4: How do I use the GCF to simplify a polynomial expression?

A4: You can use the factored expression to simplify the polynomial expression. For example, if you have a polynomial expression 6x2⋅y2−8xy2+10xy36x^2 \cdot y^2 - 8xy^2 + 10xy^3 and you want to find the GCF, you can use the factored expression 2xy2(3x−4+5xy)2xy^2(3x - 4 + 5xy) to simplify the expression.

Q5: What are some common mistakes to avoid when identifying the GCF of a polynomial expression?

A5: Some common mistakes to avoid when identifying the GCF of a polynomial expression include not factoring out the common factors from each term and not identifying the largest expression that divides each term of the polynomial without leaving a remainder.

Q6: How do I use the GCF in real-world applications?

A6: The GCF of a polynomial expression has many real-world applications, such as simplifying polynomial expressions in algebra and calculus, factoring polynomial expressions in algebra and calculus, and finding the roots of polynomial equations in algebra and calculus.

Q7: Can I use the GCF to solve polynomial equations?

A7: Yes, you can use the GCF to solve polynomial equations. By factoring out the GCF from each term, you can simplify the polynomial equation and make it easier to solve.

Q8: How do I find the GCF of a polynomial expression with multiple variables?

A8: To find the GCF of a polynomial expression with multiple variables, you need to factor out the common factors from each term. You can use the factored expression to simplify the polynomial expression and make it easier to work with.

Q9: Can I use the GCF to simplify polynomial expressions with negative coefficients?

A9: Yes, you can use the GCF to simplify polynomial expressions with negative coefficients. By factoring out the GCF from each term, you can simplify the polynomial expression and make it easier to work with.

Q10: How do I use the GCF to check if a polynomial expression is factorable?

A10: You can use the GCF to check if a polynomial expression is factorable by factoring out the GCF from each term. If the GCF is a constant, then the polynomial expression is factorable.

Conclusion

In conclusion, the GCF of a polynomial expression is an important concept in algebra and calculus. It is used to simplify polynomial expressions and make them easier to work with. By understanding the GCF and how to use it, you can simplify polynomial expressions and solve polynomial equations with ease.

Tips and Tricks

  • To find the GCF of a polynomial expression, you need to factor out the common factors from each term.
  • The GCF of a polynomial expression is the largest expression that divides each term of the polynomial without leaving a remainder.
  • You can use the factored expression to simplify the polynomial expression and make it easier to work with.

Common Mistakes

  • Not factoring out the common factors from each term.
  • Not identifying the largest expression that divides each term of the polynomial without leaving a remainder.

Real-World Applications

The GCF of a polynomial expression has many real-world applications, such as:

  • Simplifying polynomial expressions in algebra and calculus.
  • Factoring polynomial expressions in algebra and calculus.
  • Finding the roots of polynomial equations in algebra and calculus.

Conclusion

In conclusion, the GCF of a polynomial expression is an important concept in algebra and calculus. It is used to simplify polynomial expressions and make them easier to work with. By understanding the GCF and how to use it, you can simplify polynomial expressions and solve polynomial equations with ease.