Understanding Polynomials and Their Degrees

Oct 13, 2025 by windailys.com 125 views

Which Algebraic Expression is a Polynomial with a Degree of $5$ ?

Understanding Polynomials and Their Degrees

In mathematics, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The degree of a polynomial is the highest power or exponent of the variable in the expression. For example, in the polynomial $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$ , the degree is $5$ because the highest power of the variable $x$ is $5$ . In this article, we will explore which algebraic expression is a polynomial with a degree of $5$ .

What is a Polynomial?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial can be raised to various powers, and the coefficients are the numbers that multiply the variables. For example, in the polynomial $3x^2 + 2x - 4$ , the variables are $x$ and the coefficients are $3$ , $2$ , and $-4$ . The degree of this polynomial is $2$ because the highest power of the variable $x$ is $2$ .

Types of Polynomials

There are several types of polynomials, including monomials, binomials, trinomials, and polynomials with more than three terms. A monomial is a polynomial with only one term, such as $3x^2$ . A binomial is a polynomial with two terms, such as $3x^2 + 2x$ . A trinomial is a polynomial with three terms, such as $3x^2 + 2x - 4$ . A polynomial with more than three terms is called a polynomial of degree $n$ , where $n$ is the highest power of the variable.

Degree of a Polynomial

The degree of a polynomial is the highest power or exponent of the variable in the expression. For example, in the polynomial $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$ , the degree is $5$ because the highest power of the variable $x$ is $5$ . In this article, we will explore which algebraic expression is a polynomial with a degree of $5$ .

Analyzing the Options

Let's analyze the options given in the problem:

A. $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$

B. $2xy^4 + 4x^2y^3 - 6x^3y^2 - 7x^4$

C. $8y^6 + y^5 - 5xy^3 + 7x^2y^2 - x^3y - 6x^4$

D. $-6xy^5 + 5x^2y^3 - 3x^3y^2 + 2x^4y - x^5$

Option A: $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$

The degree of this polynomial is $5$ because the highest power of the variable $x$ is $5$ . This polynomial has four terms, and the highest power of the variable $x$ is $5$ . Therefore, this polynomial is a polynomial with a degree of $5$ .

Option B: $2xy^4 + 4x^2y^3 - 6x^3y^2 - 7x^4$

The degree of this polynomial is $4$ because the highest power of the variable $x$ is $4$ . This polynomial has four terms, and the highest power of the variable $x$ is $4$ . Therefore, this polynomial is not a polynomial with a degree of $5$ .

Option C: $8y^6 + y^5 - 5xy^3 + 7x^2y^2 - x^3y - 6x^4$

The degree of this polynomial is $6$ because the highest power of the variable $y$ is $6$ . This polynomial has six terms, and the highest power of the variable $y$ is $6$ . Therefore, this polynomial is not a polynomial with a degree of $5$ .

Option D: $-6xy^5 + 5x^2y^3 - 3x^3y^2 + 2x^4y - x^5$

The degree of this polynomial is $5$ because the highest power of the variable $x$ is $5$ . This polynomial has five terms, and the highest power of the variable $x$ is $5$ . Therefore, this polynomial is a polynomial with a degree of $5$ .

Conclusion

In conclusion, the algebraic expression that is a polynomial with a degree of $5$ is option A: $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$ and option D: $-6xy^5 + 5x^2y^3 - 3x^3y^2 + 2x^4y - x^5$ . Both of these polynomials have a degree of $5$ because the highest power of the variable $x$ is $5$ . Therefore, the correct answer is option A and option D.

Final Answer

The final answer is option A: $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$ and option D: $-6xy^5 + 5x^2y^3 - 3x^3y^2 + 2x^4y - x^5$ .
Q&A: Understanding Polynomials and Their Degrees

Frequently Asked Questions

In this article, we will answer some frequently asked questions about polynomials and their degrees.

Q: What is a polynomial?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial can be raised to various powers, and the coefficients are the numbers that multiply the variables.

Q: What is the degree of a polynomial?

A: The degree of a polynomial is the highest power or exponent of the variable in the expression. For example, in the polynomial $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$ , the degree is $5$ because the highest power of the variable $x$ is $5$ .

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

A: To determine the degree of a polynomial, you need to identify the highest power of the variable in the expression. You can do this by looking at the terms of the polynomial and identifying the term with the highest power of the variable.

Q: What is the difference between a polynomial and a non-polynomial expression?

A: A polynomial expression is one that consists of variables and coefficients combined using only addition, subtraction, and multiplication. A non-polynomial expression is one that includes other operations, such as division or exponentiation.

Q: Can a polynomial have a negative degree?

A: No, a polynomial cannot have a negative degree. The degree of a polynomial is always a non-negative integer.

Q: Can a polynomial have a degree of zero?

A: Yes, a polynomial can have a degree of zero. A polynomial with a degree of zero is a constant polynomial, which is a polynomial that has no variable terms.

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, you need to combine like terms and eliminate any unnecessary terms. You can do this by adding or subtracting terms with the same variable and exponent.

Q: What is the importance of understanding polynomials and their degrees?

A: Understanding polynomials and their degrees is important in mathematics and science because it allows you to analyze and solve problems involving polynomial expressions. It is also a fundamental concept in algebra and calculus.

Q: Can you provide examples of polynomials with different degrees?

A: Yes, here are some examples of polynomials with different degrees:

A polynomial with a degree of 0: $3$
A polynomial with a degree of 1: $2x + 3$
A polynomial with a degree of 2: $x^2 + 2x + 3$
A polynomial with a degree of 3: $x^3 + 2x^2 + 3x + 4$
A polynomial with a degree of 4: $x^4 + 2x^3 + 3x^2 + 4x + 5$

Q: Can you provide examples of non-polynomial expressions?

A: Yes, here are some examples of non-polynomial expressions:

A division expression: $\frac{x}{y}$
An exponentiation expression: $x^2$
A trigonometric expression: $\sin(x)$

Conclusion

In conclusion, understanding polynomials and their degrees is an important concept in mathematics and science. By understanding how to determine the degree of a polynomial and how to simplify polynomial expressions, you can analyze and solve problems involving polynomial expressions. We hope that this article has provided you with a better understanding of polynomials and their degrees.