Which Term Can Be Used In The Blank Of $36x^3 - 22x^2 - \_$ So That The Greatest Common Factor Of The Resulting Polynomial Is $2x$? Select Two Options.A. 2 B. $4xy$ C. $12x$ D. 24 E. $44y$

Introduction

In algebra, factoring polynomials is a crucial concept that helps us simplify complex expressions and solve equations. One of the key aspects of factoring polynomials is understanding the greatest common factor (GCF). In this article, we will explore the concept of GCF and how it applies to factoring polynomials. We will also examine a specific problem that requires us to find the term that can be used in the blank of a given polynomial expression so that the resulting polynomial has a GCF of $2x$.

Understanding the Greatest Common Factor

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. In the context of polynomials, the GCF is the largest polynomial that divides each of the polynomials in the set without leaving a remainder.

Factoring Polynomials with a GCF

When factoring polynomials, we often look for the GCF of the terms in the polynomial. The GCF can be a constant, a variable, or a combination of variables and constants. Once we have identified the GCF, we can factor it out of each term in the polynomial.

The Problem

The problem we are given is to find the term that can be used in the blank of the polynomial expression $36x^3 - 22x^2 - \_$ so that the resulting polynomial has a GCF of $2x$. To solve this problem, we need to analyze the given polynomial expression and determine which term will result in a GCF of $2x$.

Analyzing the Polynomial Expression

The given polynomial expression is $36x^3 - 22x^2 - \_$. To find the term that will result in a GCF of $2x$, we need to examine the coefficients of the terms in the polynomial.

• The coefficient of the first term is 36, which is a multiple of 2.
• The coefficient of the second term is -22, which is not a multiple of 2.
• To find the term that will result in a GCF of $2x$, we need to find a term that is a multiple of 2 and has a variable part of $x$.

Evaluating the Options

Let's evaluate the options given to us:

• Option A: 2. If we add 2 to the polynomial expression, we get $36x^3 - 22x^2 + 2$. The GCF of this polynomial is 2, which is not equal to $2x$.
• Option B: $4xy$. If we add $4xy$ to the polynomial expression, we get $36x^3 - 22x^2 + 4xy$. The GCF of this polynomial is $2x$, which is the desired result.
• Option C: $12x$. If we add $12x$ to the polynomial expression, we get $36x^3 - 22x^2 + 12x$. The GCF of this polynomial is $2x$, which is the desired result.
• Option D: 24. If we add 24 to the polynomial expression, we get $36x^3 - 22x^2 + 24$. The GCF of this polynomial is 2, which is not equal to $2x$.
• Option E: $44y$. If we add $44y$ to the polynomial expression, we get $36x^3 - 22x^2 + 44y$. The GCF of this polynomial is not $2x$, so this option is not correct.

Conclusion

In conclusion, the term that can be used in the blank of the polynomial expression $36x^3 - 22x^2 - \_$ so that the resulting polynomial has a GCF of $2x$ is $4xy$ or $12x$. Both of these options result in a polynomial with a GCF of $2x$, which is the desired result.

Final Answer

The final answer is:

• Option B: $4xy$
• Option C: $12x$
  Factoring Polynomials: A Q&A Guide =====================================

Introduction

In our previous article, we explored the concept of factoring polynomials and how to find the greatest common factor (GCF) of a polynomial expression. We also examined a specific problem that required us to find the term that can be used in the blank of a given polynomial expression so that the resulting polynomial has a GCF of $2x$. In this article, we will provide a Q&A guide to help you better understand the concept of factoring polynomials and how to apply it to solve problems.

Q&A Guide

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 of the terms in the polynomial without leaving a remainder.

Q: How do I find the GCF of a polynomial expression?

A: To find the GCF of a polynomial expression, you need to identify the largest expression that divides each of the terms in the polynomial without leaving a remainder. You can do this by factoring out the common factors from each term.

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

A: A common factor is a factor that divides each of the terms in a polynomial expression, but it may not be the largest factor. A greatest common factor (GCF) is the largest factor that divides each of the terms in a polynomial expression.

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

A: To factor out the GCF from a polynomial expression, you need to identify the GCF and then divide each term in the polynomial by the GCF. The result will be a polynomial expression with a GCF of 1.

Q: What is the purpose of finding the GCF of a polynomial expression?

A: The purpose of finding the GCF of a polynomial expression is to simplify the expression and make it easier to solve. By factoring out the GCF, you can reduce the complexity of the expression and make it easier to work with.

Q: Can the GCF of a polynomial expression be a variable?

A: Yes, the GCF of a polynomial expression can be a variable. For example, if the polynomial expression is $x^2 + 2x + x$, the GCF is $x$.

Q: Can the GCF of a polynomial expression be a constant?

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

Q: How do I determine if a polynomial expression has a GCF of $2x$?

A: To determine if a polynomial expression has a GCF of $2x$, you need to examine the coefficients of the terms in the polynomial. If the coefficients are multiples of 2 and have a variable part of $x$, then the polynomial expression has a GCF of $2x$.

Q: What is the term that can be used in the blank of the polynomial expression $36x^3 - 22x^2 - \_$ so that the resulting polynomial has a GCF of $2x$?

A: The term that can be used in the blank of the polynomial expression $36x^3 - 22x^2 - \_$ so that the resulting polynomial has a GCF of $2x$ is $4xy$ or $12x$.

Conclusion

In conclusion, factoring polynomials is an important concept in algebra that helps us simplify complex expressions and solve equations. By understanding the greatest common factor (GCF) of a polynomial expression, we can factor out the GCF and simplify the expression. We hope this Q&A guide has helped you better understand the concept of factoring polynomials and how to apply it to solve problems.

Final Answer

The final answer is:

• Option B: $4xy$
• Option C: $12x$