Understanding the Degree of a Polynomial

Oct 13, 2025 by windailys.com 2 views

Introduction

In algebra, a binomial is a polynomial with two terms. When we simplify a binomial expression, we are essentially combining the two terms using the rules of arithmetic. However, in this problem, we are given a partially simplified expression and asked to find the missing first term that would result in a binomial with a degree of 4 when fully simplified.

Understanding the Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in any of its terms. For example, in the polynomial $3x^2 + 2x + 1$ , the degree is 2 because the highest power of the variable $x$ is 2.

Analyzing the Given Expression

The given expression is $-5xy^3 + 9x^2y$ . To determine the missing first term, we need to understand the properties of binomials and how they are simplified.

Properties of Binomials

A binomial has two terms, and when simplified, the resulting expression is also a binomial. The general form of a binomial is $ax + b$ , where $a$ and $b$ are constants.

Simplifying the Given Expression

To simplify the given expression, we need to combine the two terms using the rules of arithmetic. However, we are not given the second term, so we will assume it is $cx^2y$ .

Combining Like Terms

When combining like terms, we add or subtract the coefficients of the terms with the same variable and exponent.

Example

Suppose the second term is $3x^2y$ . When we combine the two terms, we get:

-5xy^3 + 3x^2y

However, this expression is not a binomial with a degree of 4. To make it a binomial with a degree of 4, we need to add a term with a higher power of the variable.

Finding the Missing Term

To find the missing term, we need to analyze the given expression and determine what term would result in a binomial with a degree of 4 when fully simplified.

Option A: 0

If the missing term is 0, the expression would be $-5xy^3$ . However, this expression is not a binomial with a degree of 4.

Option B: $5xy^3$

If the missing term is $5xy^3$ , the expression would be $-5xy^3 + 5xy^3 = 0$ . However, this expression is not a binomial with a degree of 4.

Option C: $9x^2y$

If the missing term is $9x^2y$ , the expression would be $-5xy^3 + 9x^2y$ . However, this expression is not a binomial with a degree of 4.

Conclusion

Based on the analysis, we can conclude that the missing term is not any of the options given. However, we can try to find the missing term by analyzing the given expression and determining what term would result in a binomial with a degree of 4 when fully simplified.

The Correct Answer

After analyzing the given expression, we can conclude that the missing term is $-9x^2y$ . When we add this term to the given expression, we get:

-5xy^3 - 9x^2y + 9x^2y

However, this expression is not a binomial with a degree of 4. We need to add another term to make it a binomial with a degree of 4.

The Final Answer

After analyzing the given expression, we can conclude that the missing term is $-9x^2y + 5xy^3$ . When we add this term to the given expression, we get:

-5xy^3 - 9x^2y + 5xy^3 + 9x^2y

This expression is a binomial with a degree of 4.

The Final Answer is:

$The final answer is $\boxed{-9x^2y + 5xy^3}$.<br/> **Frequently Asked Questions (FAQs)**$

Q: What is a binomial?

A: A binomial is a polynomial with two terms. When we simplify a binomial expression, we are essentially combining the two terms using the rules of arithmetic.

Q: What is the degree of a polynomial?

A: The degree of a polynomial is the highest power of the variable in any of its terms. For example, in the polynomial $3x^2 + 2x + 1$ , the degree is 2 because the highest power of the variable $x$ is 2.

Q: How do I simplify a binomial expression?

A: To simplify a binomial expression, we need to combine the two terms using the rules of arithmetic. We add or subtract the coefficients of the terms with the same variable and exponent.

Q: What is the missing term in the expression $-5xy^3 + 9x^2y$ ?

A: The missing term in the expression $-5xy^3 + 9x^2y$ is $-9x^2y + 5xy^3$ . When we add this term to the given expression, we get a binomial with a degree of 4.

Q: How do I determine the missing term in a binomial expression?

A: To determine the missing term in a binomial expression, we need to analyze the given expression and determine what term would result in a binomial with a degree of 4 when fully simplified.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{-9x^2y + 5xy^3}$ .

Q: Can you explain the concept of like terms?

A: Yes, like terms are terms that have the same variable and exponent. When we combine like terms, we add or subtract the coefficients of the terms with the same variable and exponent.

Q: How do I combine like terms?

A: To combine like terms, we add or subtract the coefficients of the terms with the same variable and exponent.

Q: What is the importance of understanding binomial expressions?

A: Understanding binomial expressions is important because it helps us to simplify complex algebraic expressions and solve problems in mathematics and science.

Q: Can you provide examples of binomial expressions?

A: Yes, here are some examples of binomial expressions:

$2x + 3$
$x^2 + 4x$
$3y^2 - 2y$

Q: How do I simplify binomial expressions with variables?

A: To simplify binomial expressions with variables, we need to combine the two terms using the rules of arithmetic. We add or subtract the coefficients of the terms with the same variable and exponent.

Q: What is the difference between a binomial and a polynomial?

A: A binomial is a polynomial with two terms, while a polynomial is a general term that can have any number of terms.

Q: Can you explain the concept of degree of a polynomial?

A: Yes, the degree of a polynomial is the highest power of the variable in any of its terms. For example, in the polynomial $3x^2 + 2x + 1$ , the degree is 2 because the highest power of the variable $x$ is 2.

Q: How do I determine the degree of a polynomial?

A: To determine the degree of a polynomial, we need to identify the highest power of the variable in any of its terms.

Q: What is the importance of understanding the degree of a polynomial?

A: Understanding the degree of a polynomial is important because it helps us to determine the complexity of the polynomial and to simplify it using the rules of arithmetic.