Factor Out The Greatest Common Factor From The Following Polynomial: 16 A 3 + 4 A 2 + 1 16a^3 + 4a^2 + 1 16 A 3 + 4 A 2 + 1 Select The Correct Choice Below:A. 16 A 3 + 4 A 2 + 1 = 16a^3 + 4a^2 + 1 = 16 A 3 + 4 A 2 + 1 = { \square$}$ (Factor Completely.)B. The Polynomial Has No Common Factor Other Than 1.

Introduction

Factoring out the greatest common factor (GCF) from a polynomial is an essential skill in algebra. It involves identifying the common factor among the terms of a polynomial and expressing the polynomial as a product of the GCF and a new polynomial. In this article, we will learn how to factor out the GCF from the given polynomial: $16a^3 + 4a^2 + 1$.

Understanding the Greatest Common Factor

The greatest common factor (GCF) of a set of numbers is the largest number that divides each of the numbers in the set without leaving a remainder. In the context of polynomials, the GCF is the common factor among the terms of the polynomial. To factor out the GCF, we need to identify the common factor and express the polynomial as a product of the GCF and a new polynomial.

Step 1: Identify the Common Factor

To factor out the GCF, we need to identify the common factor among the terms of the polynomial. In this case, the polynomial is $16a^3 + 4a^2 + 1$. We can see that the common factor among the terms is $1$. However, we can also see that the terms $16a^3$ and $4a^2$ have a common factor of $4a^2$.

Step 2: Factor Out the Common Factor

Now that we have identified the common factor, we can factor it out from the polynomial. To do this, we need to divide each term of the polynomial by the common factor. In this case, we can divide each term by $4a^2$.

$16a^3 + 4a^2 + 1 = 4a^2(4a) + 4a^2(1) + 1$

$16a^3 + 4a^2 + 1 = 16a^3 + 4a^2 + 1$

Step 3: Simplify the Expression

Now that we have factored out the common factor, we can simplify the expression. In this case, we can see that the expression is already simplified.

$16a^3 + 4a^2 + 1 = 4a^2(4a) + 4a^2(1) + 1$

$16a^3 + 4a^2 + 1 = 16a^3 + 4a^2 + 1$

Conclusion

In conclusion, we have learned how to factor out the greatest common factor from the given polynomial: $16a^3 + 4a^2 + 1$. We identified the common factor as $1$ and factored it out from the polynomial. We then simplified the expression to get the final result.

Answer

The correct answer is:

B. The polynomial has no common factor other than 1.

Discussion

The discussion category for this problem is mathematics. The problem involves factoring out the greatest common factor from a polynomial, which is a fundamental concept in algebra. The solution requires the student to identify the common factor and express the polynomial as a product of the GCF and a new polynomial.

Related Topics

Some related topics to this problem include:

• Factoring out the greatest common factor from a polynomial
• Identifying the common factor among the terms of a polynomial
• Expressing a polynomial as a product of the GCF and a new polynomial
• Simplifying an expression by factoring out the common factor

Example Problems

Some example problems that involve factoring out the greatest common factor from a polynomial include:

• Factor out the greatest common factor from the polynomial: $9x^3 + 3x^2 + 1$
• Factor out the greatest common factor from the polynomial: $12y^3 + 4y^2 + 1$
• Factor out the greatest common factor from the polynomial: $15z^3 + 5z^2 + 1$

Practice Problems

Some practice problems that involve factoring out the greatest common factor from a polynomial include:

• Factor out the greatest common factor from the polynomial: $18a^3 + 6a^2 + 1$
• Factor out the greatest common factor from the polynomial: $24b^3 + 8b^2 + 1$
• Factor out the greatest common factor from the polynomial: $30c^3 + 10c^2 + 1$

Glossary

Some key terms that are related to this problem include:

• Greatest common factor (GCF): the largest number that divides each of the numbers in a set without leaving a remainder
• Common factor: a factor that is common to all the terms of a polynomial
• Factoring out the GCF: expressing a polynomial as a product of the GCF and a new polynomial
• Simplifying an expression: expressing an expression in its simplest form by factoring out the common factor.
  Q&A: Factoring Out the Greatest Common Factor =====================================================

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

A: The greatest common factor (GCF) of a polynomial is the largest number that divides each of the terms of the polynomial without leaving a remainder.

Q: How do I identify the common factor among the terms of a polynomial?

A: To identify the common factor, you need to look for the largest number that divides each of the terms of the polynomial without leaving a remainder. You can also use the following steps:

1. List the terms of the polynomial.
2. Identify the common factor among the terms.
3. Check if the common factor divides each of the terms without leaving a remainder.

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

A: To factor out the greatest common factor, you need to follow these steps:

1. Identify the common factor among the terms of the polynomial.
2. Divide each term of the polynomial by the common factor.
3. Express the polynomial as a product of the GCF and a new polynomial.

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 a new polynomial, while factoring a polynomial involves expressing it as a product of its roots or factors.

Q: Can a polynomial have more than one common factor?

A: Yes, a polynomial can have more than one common factor. However, the greatest common factor is the largest number that divides each of the terms of the polynomial without leaving a remainder.

Q: How do I simplify an expression by factoring out the common factor?

A: To simplify an expression by factoring out the common factor, you need to follow these steps:

1. Identify the common factor among the terms of the expression.
2. Divide each term of the expression by the common factor.
3. Express the expression as a product of the GCF and a new expression.

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:

• Not identifying the common factor correctly.
• Not dividing each term of the polynomial by the common factor.
• Not expressing the polynomial as a product of the GCF and a new polynomial.

Q: How do I check if I have factored out the GCF correctly?

A: To check if you have factored out the GCF correctly, you need to follow these steps:

1. Multiply the GCF and the new polynomial to get the original polynomial.
2. Check if the product is equal to the original polynomial.
3. If the product is equal to the original polynomial, then you have factored out the GCF correctly.

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

A: Some real-world applications of factoring out the GCF include:

• Simplifying complex expressions in physics and engineering.
• Factoring out the GCF in algebraic equations.
• Using the GCF to solve systems of equations.

Q: Can I use technology to help me factor out the GCF?

A: Yes, you can use technology to help you factor out the GCF. Some popular tools include:

• Graphing calculators.
• Algebra software.
• Online factoring tools.

Q: How do I choose the right tool to help me factor out the GCF?

A: To choose the right tool, you need to consider the following factors:

• The complexity of the polynomial.
• The level of accuracy required.
• The availability of the tool.

Q: What are some common mistakes to avoid when using technology to factor out the GCF?

A: Some common mistakes to avoid when using technology to factor out the GCF include:

• Not using the correct tool for the job.
• Not entering the polynomial correctly.
• Not checking the accuracy of the result.