What Is The Greatest Common Factor Of $8xy^5 - 16x 2y 3 + 20x 4y 4 ? ? ? A. $4x 4y 5$ B. $ 2 X Y 3 2xy^3 2 X Y 3 [/tex] C. $4xy^3$ D. $8xy^5$

Introduction to Greatest Common Factor (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 the terms in the polynomial. Finding the GCF of a polynomial expression is an essential skill in algebra, as it helps us simplify complex expressions and solve equations.

Understanding the Polynomial Expression

The given polynomial expression is $8xy^5 - 16x^2y3 + 20x^4y4$. To find the GCF, we need to identify the common factors of the terms in the expression. The terms are $8xy^5$, $-16x^2y3$, and $20x^4y4$.

Factoring Out the Greatest Common Factor

To find the GCF, we need to factor out the common factors from each term. The common factors are $4$, $x$, and $y^3$. We can factor out these common factors from each term as follows:

$$8xy^5 = 4 \cdot 2xy^5$$

$$-16x^2y^3 = -4 \cdot 4x^2y^3$$

$$20x^4y^4 = 4 \cdot 5x^4y^4$$

Identifying the Greatest Common Factor

Now that we have factored out the common factors from each term, we can identify the GCF. The GCF is the product of the common factors, which is $4xy^3$.

Conclusion

In conclusion, the greatest common factor of the polynomial expression $8xy^5 - 16x^2y3 + 20x^4y4$ is $4xy^3$. This is the largest expression that divides each term of the polynomial without leaving a remainder.

Comparison with Other Options

Let's compare our answer with the other options:

• Option A: $4x^4y5$ is not the GCF, as it does not divide each term of the polynomial without leaving a remainder.
• Option B: $2xy^3$ is not the GCF, as it is not the largest expression that divides each term of the polynomial without leaving a remainder.
• Option D: $8xy^5$ is not the GCF, as it does not divide each term of the polynomial without leaving a remainder.

Final Answer

The final answer is $\boxed{C. 4xy^3}$.

Frequently Asked Questions

• What is the greatest common factor of a polynomial expression? The greatest common factor of a polynomial expression is the largest expression that divides each term of the polynomial without leaving a remainder.
• How do I find the greatest common factor of a polynomial expression? To find the greatest common factor of a polynomial expression, you need to identify the common factors of the terms in the expression and factor them out.
• What is the product of the common factors? The product of the common factors is the greatest common factor of the polynomial expression.

Step-by-Step Solution

1. Identify the common factors of the terms in the polynomial expression.
2. Factor out the common factors from each term.
3. Identify the greatest common factor by taking the product of the common factors.

Example Problems

• Find the greatest common factor of the polynomial expression $12x^2y3 - 18x^3y4 + 24x^4y5$.
• Find the greatest common factor of the polynomial expression $15x^2y2 - 20x^3y3 + 25x^4y4$.

Tips and Tricks

• Make sure to identify the common factors of the terms in the polynomial expression.
• Factor out the common factors from each term.
• Take the product of the common factors to find the greatest common factor.

Conclusion

In conclusion, finding the greatest common factor of a polynomial expression is an essential skill in algebra. By identifying the common factors of the terms in the expression and factoring them out, we can find the greatest common factor. The product of the common factors is the greatest common factor of the polynomial expression.

Q: What is the greatest common factor (GCF) 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 greatest common factor of a polynomial expression, you need to identify the common factors of the terms in the expression and factor them out.

Q: What are the common factors of a polynomial expression?

A: The common factors of a polynomial expression are the factors that are common to all the terms in the expression.

Q: How do I identify the common factors of a polynomial expression?

A: To identify the common factors of a polynomial expression, you need to look for the factors that are present in all the terms of the expression.

Q: What is the product of the common factors?

A: The product of the common factors is the greatest common factor of the polynomial expression.

Q: Can the greatest common factor be a constant?

A: Yes, the greatest common factor can be a constant. For example, if the polynomial expression is $2x^2 + 4x^2 + 6x^2$, the greatest common factor is $2$.

Q: Can the greatest common factor be a variable?

A: Yes, the greatest common factor can be a variable. For example, if the polynomial expression is $x^2 + 2x^2 + 3x^2$, the greatest common factor is $x^2$.

Q: Can the greatest common factor be a product of a constant and a variable?

A: Yes, the greatest common factor can be a product of a constant and a variable. For example, if the polynomial expression is $2x^2 + 4x^2 + 6x^2$, the greatest common factor is $2x^2$.

Q: How do I simplify a polynomial expression using the greatest common factor?

A: To simplify a polynomial expression using the greatest common factor, you need to divide each term of the expression by the greatest common factor.

Q: What is the difference between the greatest common factor and the least common multiple?

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

Q: Can the greatest common factor be used to solve equations?

A: Yes, the greatest common factor can be used to solve equations. By factoring out the greatest common factor from each term of the equation, you can simplify the equation and solve for the variable.

Q: Can the greatest common factor be used to simplify expressions?

A: Yes, the greatest common factor can be used to simplify expressions. By factoring out the greatest common factor from each term of the expression, you can simplify the expression and make it easier to work with.

Q: What are some common mistakes to avoid when finding the greatest common factor?

A: Some common mistakes to avoid when finding the greatest common factor include:

• Not identifying the common factors of the terms in the expression
• Not factoring out the common factors from each term
• Not taking the product of the common factors to find the greatest common factor

Q: How do I check my answer for the greatest common factor?

A: To check your answer for the greatest common factor, you need to divide each term of the expression by the greatest common factor and make sure that the remainder is zero.

Q: What are some real-world applications of the greatest common factor?

A: Some real-world applications of the greatest common factor include:

• Simplifying complex expressions in physics and engineering
• Solving equations in finance and economics
• Finding the greatest common divisor of two numbers in computer science

Q: Can the greatest common factor be used to solve problems in other areas of mathematics?

A: Yes, the greatest common factor can be used to solve problems in other areas of mathematics, such as algebra, geometry, and trigonometry.

Q: What are some tips for finding the greatest common factor?

A: Some tips for finding the greatest common factor include:

• Identifying the common factors of the terms in the expression
• Factoring out the common factors from each term
• Taking the product of the common factors to find the greatest common factor
• Checking your answer by dividing each term of the expression by the greatest common factor.