The Polynomial in Question

Oct 13, 2025 by talynt.org 107 views

Introduction

In algebra, a factor of a polynomial is an expression that divides the polynomial without leaving a remainder. Factoring polynomials is an essential skill in mathematics, as it allows us to simplify complex expressions, identify common factors, and solve equations. In this article, we will explore the factors of a given polynomial and determine which of the options is a factor.

The Polynomial in Question

The polynomial we will be analyzing is $$500x^3 + 108y^{18}$$. This polynomial consists of two terms, each with a different variable and exponent. To determine the factors of this polynomial, we need to examine each term separately.

Factoring the First Term

The first term of the polynomial is $$500x^3$$. This term can be factored as follows:

500x^3 = 100 \cdot 5x^3 = 100x^2 \cdot 5x

We can see that the first term can be factored into two parts: $$100x^2$$ and $$5x$$.

Factoring the Second Term

The second term of the polynomial is $$108y^{18}$$. This term can be factored as follows:

108y^{18} = 27 \cdot 4 \cdot 3^3 \cdot y^{18} = 27 \cdot 4 \cdot (3y^6)^3 = 27 \cdot 4 \cdot 3^3 \cdot y^{18}

We can see that the second term can be factored into three parts: $$27$$, $$4$$, and $$(3y⁶⁾3$$.

Examining the Options

Now that we have factored the polynomial, let's examine the options:

A. 6 B. $$5x + 3y^6$$ C. $$25x^2 + 15xy^6 + 9y^{12}$$ D. All of the above

Option A: 6

Option A is the number 6. We can see that 6 is not a factor of the polynomial, as it does not divide the polynomial without leaving a remainder.

Option B: $$5x + 3y^6$$

Option B is the expression $$5x + 3y^6$$. We can see that this expression is a factor of the polynomial, as it divides the polynomial without leaving a remainder.

Option C: $$25x^2 + 15xy^6 + 9y^{12}$$

Option C is the expression $$25x^2 + 15xy^6 + 9y^{12}$$. We can see that this expression is also a factor of the polynomial, as it divides the polynomial without leaving a remainder.

Conclusion

In conclusion, we have analyzed the factors of the polynomial $$500x^3 + 108y^{18}$$. We have determined that the expression $$5x + 3y^6$$ is a factor of the polynomial, and that the expression $$25x^2 + 15xy^6 + 9y^{12}$$ is also a factor of the polynomial. Therefore, the correct answer is:

The Final Answer

Q&A: Factors of Polynomials

Q: What is a factor of a polynomial?

A: A factor of a polynomial is an expression that divides the polynomial without leaving a remainder.

Q: How do you determine the factors of a polynomial?

A: To determine the factors of a polynomial, you need to examine each term separately and look for common factors. You can also use factoring techniques such as factoring out the greatest common factor (GCF) or using the distributive property.

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

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

Q: How do you factor out the GCF?

A: To factor out the GCF, you need to identify the GCF and divide each term of the polynomial by the GCF.

Q: What is the distributive property?

A: The distributive property is a factoring technique that allows you to multiply a single term by each term in a polynomial.

Q: How do you use the distributive property?

A: To use the distributive property, you need to multiply a single term by each term in a polynomial and then combine like terms.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent.

Q: How do you combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms.

Q: What is the difference between a factor and a multiple?

A: A factor is an expression that divides a polynomial without leaving a remainder, while a multiple is an expression that is obtained by multiplying a polynomial by a constant.

Q: How do you determine if an expression is a factor or a multiple?

A: To determine if an expression is a factor or a multiple, you need to check if the expression divides the polynomial without leaving a remainder.

Q: What are some common factoring techniques?

A: Some common factoring techniques include factoring out the GCF, using the distributive property, and factoring by grouping.

Q: How do you factor by grouping?

A: To factor by grouping, you need to group the terms of a polynomial into two or more groups and then factor out the GCF from each group.

Q: What are some common mistakes to avoid when factoring polynomials?

A: Some common mistakes to avoid when factoring polynomials include forgetting to factor out the GCF, not combining like terms, and not checking if an expression is a factor or a multiple.

Conclusion

In conclusion, factoring polynomials is an essential skill in mathematics that requires a deep understanding of algebraic concepts. By understanding the factors of a polynomial, you can simplify complex expressions, identify common factors, and solve equations. We hope this article has provided you with a comprehensive understanding of factors of polynomials and has helped you to improve your factoring skills.

Additional Resources

Khan Academy: Factoring Polynomials
Mathway: Factoring Polynomials
Wolfram Alpha: Factoring Polynomials

Final Tips

Practice factoring polynomials regularly to improve your skills.
Use online resources such as Khan Academy, Mathway, and Wolfram Alpha to help you with factoring polynomials.
Don't be afraid to ask for help if you are struggling with factoring polynomials.