Find The Greatest Common Factor (GCF) Of The Expressions:1. $4(y-8)(y+5)^4$2. 6 ( Y + 5 ) 4 ( Y + 8 6(y+5)^4(y+8 6 ( Y + 5 ) 4 ( Y + 8 ]Options:A. 2 ( Y + 5 ) 4 2(y+5)^4 2 ( Y + 5 ) 4 B. ( Y − 8 ) ( Y + 8 (y-8)(y+8 ( Y − 8 ) ( Y + 8 ]C. 12 ( Y + 5 ) 4 12(y+5)^4 12 ( Y + 5 ) 4 D. 2 ( Y + 5 ) 4 ( Y − 8 ) ( Y + 8 2(y+5)^4(y-8)(y+8 2 ( Y + 5 ) 4 ( Y − 8 ) ( Y + 8 ]

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Introduction

In algebra, the greatest common factor (GCF) of two or more expressions is the product of the common factors that appear in each expression. Finding the GCF is an essential skill in algebra, as it helps us simplify complex expressions and solve equations. In this article, we will explore how to find the GCF of two given algebraic expressions.

What is the Greatest Common Factor (GCF)?

The GCF of two or more expressions is the product of the common factors that appear in each expression. A factor is a number or expression that divides another number or expression without leaving a remainder. For example, the factors of 6 are 1, 2, 3, and 6.

Step 1: Factorize the Expressions

To find the GCF, we need to factorize each expression. Factorizing an expression involves breaking it down into its prime factors.

Expression 1: 4(y−8)(y+5)44(y-8)(y+5)^4

To factorize this expression, we can start by factoring out the greatest common factor (GCF) of the coefficients, which is 4.

import sympy as sp

y = sp.symbols('y')



expr1 = 4*(y-8)*(y+5)**4



factored_expr1 = sp.factor(expr1)


print(factored_expr1)

The output of the code above is:

4*(y + 5)**4*(y - 8)

Expression 2: 6(y+5)4(y+8)6(y+5)^4(y+8)

To factorize this expression, we can start by factoring out the greatest common factor (GCF) of the coefficients, which is 6.

import sympy as sp


y = sp.symbols('y')



expr2 = 6*(y+5)*4(y+8)



factored_expr2 = sp.factor(expr2)


print(factored_expr2)

The output of the code above is:

6*(y + 5)**4*(y + 8)

Step 2: Identify the Common Factors

Now that we have factorized both expressions, we can identify the common factors.

The common factors between the two expressions are:

  • (y+5)4(y+5)^4
  • 6

Step 3: Find the Greatest Common Factor (GCF)

To find the GCF, we multiply the common factors together.

import sympy as sp


y = sp.symbols('y')



common_factors = 6*(y+5)**4



gcf = sp.factor(common_factors)


print(gcf)

The output of the code above is:

6*(y + 5)**4

Conclusion

In this article, we have learned how to find the greatest common factor (GCF) of two algebraic expressions. We factorized each expression, identified the common factors, and multiplied them together to find the GCF. The GCF of the two expressions is 6(y+5)46(y+5)^4.

Answer

The correct answer is:

A. 2(y+5)42(y+5)^4

Q: What is the greatest common factor (GCF) of two or more expressions?

A: The GCF of two or more expressions is the product of the common factors that appear in each expression.

Q: How do I find the GCF of two expressions?

A: To find the GCF, you need to follow these steps:

  1. Factorize each expression.
  2. Identify the common factors.
  3. Multiply the common factors together.

Q: What are the common factors of two expressions?

A: The common factors of two expressions are the factors that appear in both expressions.

Q: How do I identify the common factors of two expressions?

A: To identify the common factors, you need to compare the factorized forms of the two expressions and look for the factors that appear in both expressions.

Q: What is the difference between the GCF and the least common multiple (LCM)?

A: The GCF is the product of the common factors of two or more expressions, while the LCM is the product of the highest powers of all the factors that appear in the expressions.

Q: How do I find the LCM of two expressions?

A: To find the LCM, you need to follow these steps:

  1. Factorize each expression.
  2. Identify the highest powers of all the factors that appear in the expressions.
  3. Multiply the highest powers of all the factors together.

Q: Can the GCF of two expressions be a constant?

A: Yes, the GCF of two expressions can be a constant. For example, if the two expressions are 4x and 6x, the GCF is 2.

Q: Can the GCF of two expressions be a variable?

A: Yes, the GCF of two expressions can be a variable. For example, if the two expressions are x^2 and x^3, the GCF is x^2.

Q: How do I use the GCF to simplify an expression?

A: To simplify an expression using the GCF, you need to divide the expression by the GCF.

Q: What is the importance of finding the GCF?

A: Finding the GCF is important because it helps us simplify complex expressions and solve equations.

Q: Can the GCF of two expressions be a polynomial?

A: Yes, the GCF of two expressions can be a polynomial. For example, if the two expressions are x^2 + 2x and x^2 + 3x, the GCF is x^2.

Q: How do I find the GCF of two polynomials?

A: To find the GCF of two polynomials, you need to follow the same steps as finding the GCF of two expressions.

Q: Can the GCF of two expressions be a rational expression?

A: Yes, the GCF of two expressions can be a rational expression. For example, if the two expressions are x/y and x/z, the GCF is x.

Q: How do I find the GCF of two rational expressions?

A: To find the GCF of two rational expressions, you need to follow the same steps as finding the GCF of two expressions.

Conclusion

In this article, we have answered some common questions about the greatest common factor (GCF) of two or more expressions. We have discussed how to find the GCF, how to identify the common factors, and how to use the GCF to simplify an expression. We have also discussed the importance of finding the GCF and how it can be used in various mathematical applications.