Arpitha Factors Out The Greatest Common Factor, $9n$, From The Terms Of The Polynomial Shown:$162m^3n^4 + 45n = 9n(\,\, - \,\, + 5$\]What Is The Missing Term In The Factored Expression?A. $16m^3n^4$ B. $16m^3n^3$

Understanding the Greatest Common Factor (GCF)

In mathematics, the greatest common factor (GCF) of a set of numbers or expressions is the largest expression that divides each of the numbers or expressions without leaving a remainder. When factoring out the GCF from a polynomial, we are essentially expressing the polynomial as a product of the GCF and another expression.

The Given Polynomial

The given polynomial is $162m^3n^4 + 45n$. We are asked to factor out the greatest common factor, $9n$, from the terms of the polynomial.

Factoring Out the GCF

To factor out the GCF, we need to identify the common factors between the two terms. In this case, the common factors are $9$ and $n$. We can factor out $9n$ from both terms as follows:

$$162m^3n^4 + 45n = 9n(18m^3n^3 + 5)$$

Identifying the Missing Term

The factored expression is $9n(18m^3n^3 + 5)$. We are asked to identify the missing term in the factored expression. The missing term is the expression that is being multiplied by the GCF, $9n$. In this case, the missing term is $18m^3n^3$.

Conclusion

In conclusion, the missing term in the factored expression is $18m^3n^3$. This is because the GCF, $9n$, has been factored out of the polynomial, leaving the expression $18m^3n^3 + 5$ as the missing term.

Answer

The correct answer is B. $16m^3n^3$ is incorrect, the correct answer is $18m^3n^3$.

Step-by-Step Solution

1. Identify the GCF of the two terms: $9n$
2. Factor out the GCF from both terms: $9n(18m^3n^3 + 5)$
3. Identify the missing term in the factored expression: $18m^3n^3$

Common Mistakes

• Failing to identify the GCF of the two terms
• Failing to factor out the GCF from both terms
• Identifying the wrong missing term in the factored expression

Tips and Tricks

• Make sure to identify the GCF of the two terms before factoring it out.
• Factor out the GCF from both terms to ensure that the expression is factored correctly.
• Identify the missing term in the factored expression by looking at the expression that is being multiplied by the GCF.
  Factoring Out the Greatest Common Factor in Polynomials: Q&A ===========================================================

Q: What is the greatest common factor (GCF) of a set of numbers or expressions?

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

Q: How do I identify the GCF of two terms?

A: To identify the GCF of two terms, you need to find the largest expression that divides both terms without leaving a remainder. You can do this by listing the factors of each term and finding the common factors.

Q: What is the difference between factoring out the GCF and factoring a polynomial?

A: Factoring out the GCF involves expressing a polynomial as a product of the GCF and another expression. Factoring a polynomial involves expressing it as a product of simpler polynomials.

Q: How do I factor out the GCF from a polynomial?

A: To factor out the GCF from a polynomial, you need to identify the common factors between the terms and factor them out. You can do this by dividing each term by the GCF and simplifying the expression.

Q: What is the missing term in the factored expression $9n(18m^3n^3 + 5)$?

A: The missing term in the factored expression $9n(18m^3n^3 + 5)$ is $18m^3n^3$.

Q: Why is it important to identify the GCF of a set of numbers or expressions?

A: Identifying the GCF of a set of numbers or expressions is important because it allows you to simplify the expression and make it easier to work with.

Q: Can you give an example of a polynomial that can be factored out by the GCF?

A: Yes, an example of a polynomial that can be factored out by the GCF is $12x^2 + 18x$. The GCF of this polynomial is $6x$, and factoring it out gives $6x(2x + 3)$.

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

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

• Failing to identify the GCF of the two terms
• Failing to factor out the GCF from both terms
• Identifying the wrong missing term in the factored expression

Q: How can I practice factoring out the GCF?

A: You can practice factoring out the GCF by working through examples and exercises in your textbook or online resources. You can also try factoring out the GCF from your own expressions and polynomials.

Q: What are some real-world applications of factoring out the GCF?

A: Factoring out the GCF has many real-world applications, including:

• Simplifying complex expressions in physics and engineering
• Factoring polynomials in algebra and calculus
• Solving systems of equations in mathematics and science

Q: Can you give an example of a real-world application of factoring out the GCF?

A: Yes, an example of a real-world application of factoring out the GCF is in physics, where it is used to simplify complex expressions and solve problems involving motion and energy.