Factor Out The Greatest Common Factor.${ 16m^{10} - 48m^9 }$

Understanding the Greatest Common Factor (GCF)

The greatest common factor (GCF) is a mathematical concept that plays a crucial role in algebraic expressions. It is the largest expression that divides two or more numbers or expressions without leaving a remainder. In the context of the given problem, we are required to factor out the greatest common factor from the expression $16m^{10} - 48m^9$.

Identifying the Greatest Common Factor

To factor out the greatest common factor, we need to identify the common factors between the two terms in the expression. The first step is to look for the common factors in the coefficients and the variables. In this case, the coefficients are 16 and 48, and the variables are $m^{10}$ and $m^9$.

Breaking Down the Coefficients

The coefficients 16 and 48 can be broken down into their prime factors. The prime factorization of 16 is $2^4$, and the prime factorization of 48 is $2^4 \cdot 3$. By comparing the prime factorizations, we can see that the greatest common factor of the coefficients is $2^4$, which is equal to 16.

Breaking Down the Variables

The variables $m^{10}$ and $m^9$ can be broken down by subtracting the exponents. The difference between the exponents is 1, which means that the greatest common factor of the variables is $m^1$, or simply $m$.

Factoring Out the Greatest Common Factor

Now that we have identified the greatest common factor of the coefficients and the variables, we can factor it out from the expression. The greatest common factor is 16, and the variables are $m^{10}$ and $m^9$. To factor out the greatest common factor, we need to divide each term by 16 and multiply the result by $m$.

Step 1: Divide Each Term by 16

The first term is $16m^{10}$, and the second term is $-48m^9$. To divide each term by 16, we need to divide the coefficients by 16 and divide the variables by $m^{10}$ and $m^9$.

Step 2: Multiply the Result by $m$

After dividing each term by 16, we get $m^{10}$ and $-3m^9$. To multiply the result by $m$, we need to multiply each term by $m$.

Factored Form

The factored form of the expression is $16m(m^{9} - 3m^8)$.

Conclusion

In conclusion, we have successfully factored out the greatest common factor from the expression $16m^{10} - 48m^9$. The greatest common factor is 16, and the factored form of the expression is $16m(m^{9} - 3m^8)$. This demonstrates the importance of identifying the greatest common factor in algebraic expressions and how it can be used to simplify complex expressions.

Real-World Applications

The concept of greatest common factor has numerous real-world applications in various fields, including mathematics, science, and engineering. For example, in physics, the greatest common factor is used to calculate the frequency of a wave, while in engineering, it is used to design and optimize systems.

Common Mistakes to Avoid

When factoring out the greatest common factor, there are several common mistakes to avoid. These include:

• Not identifying the greatest common factor: This is the most common mistake when factoring out the greatest common factor. It is essential to identify the greatest common factor correctly to avoid errors.
• Not factoring out the greatest common factor completely: This can lead to incorrect results and make it difficult to simplify the expression.
• Not checking for common factors: This can lead to missing common factors and incorrect results.

Tips and Tricks

When factoring out the greatest common factor, there are several tips and tricks to keep in mind. These include:

• Use prime factorization: Prime factorization is a powerful tool for identifying the greatest common factor.
• Look for common factors: Look for common factors in the coefficients and the variables.
• Check for common factors: Check for common factors in the coefficients and the variables.
• Use the distributive property: The distributive property can be used to factor out the greatest common factor.

Practice Problems

To practice factoring out the greatest common factor, try the following problems:

• Problem 1: Factor out the greatest common factor from the expression $24x^3 - 36x^2$.
• Problem 2: Factor out the greatest common factor from the expression $48y^4 - 32y^3$.
• Problem 3: Factor out the greatest common factor from the expression $60z^5 - 40z^4$.

Conclusion

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest expression that divides two or more numbers or expressions without leaving a remainder.

Q: How do I identify the greatest common factor?

A: To identify the greatest common factor, you need to look for the common factors in the coefficients and the variables. You can use prime factorization to break down the coefficients and the variables.

Q: What is prime factorization?

A: Prime factorization is the process of breaking down a number or expression into its prime factors. For example, the prime factorization of 16 is $2^4$, and the prime factorization of 48 is $2^4 \cdot 3$.

Q: How do I factor out the greatest common factor?

A: To factor out the greatest common factor, you need to divide each term by the greatest common factor and multiply the result by the greatest common factor.

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

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

• Not identifying the greatest common factor
• Not factoring out the greatest common factor completely
• Not checking for common factors

Q: How do I check for common factors?

A: To check for common factors, you need to look for common factors in the coefficients and the variables. You can use prime factorization to break down the coefficients and the variables.

Q: What are some tips and tricks for factoring out the greatest common factor?

A: Some tips and tricks for factoring out the greatest common factor include:

• Use prime factorization
• Look for common factors
• Check for common factors
• Use the distributive property

Q: Can you provide some practice problems for factoring out the greatest common factor?

A: Yes, here are some practice problems for factoring out the greatest common factor:

• Problem 1: Factor out the greatest common factor from the expression $24x^3 - 36x^2$.
• Problem 2: Factor out the greatest common factor from the expression $48y^4 - 32y^3$.
• Problem 3: Factor out the greatest common factor from the expression $60z^5 - 40z^4$.

Q: How do I know if I have factored out the greatest common factor correctly?

A: To know if you have factored out the greatest common factor correctly, you need to check if the expression is simplified and if the greatest common factor is factored out completely.

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

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

• Calculating the frequency of a wave in physics
• Designing and optimizing systems in engineering
• Simplifying complex expressions in mathematics

Q: Can you provide some additional resources for learning about factoring out the greatest common factor?

A: Yes, here are some additional resources for learning about factoring out the greatest common factor:

• Online tutorials and videos
• Math textbooks and workbooks
• Online math communities and forums

Conclusion

In conclusion, factoring out the greatest common factor is a crucial concept in algebraic expressions. By understanding the greatest common factor and how to factor it out, you can simplify complex expressions and apply it to real-world problems. We hope this Q&A article has provided you with a better understanding of factoring out the greatest common factor and has helped you to master this concept.