Breaking Down the Polynomial

Oct 13, 2025 by leolavanderia.com.br 125 views

Introduction

Polynomials are algebraic expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication. In this article, we will focus on identifying the coefficient and degree of each term in a given polynomial expression. The given polynomial is $x^8 y - 3xy + 8x^4 - 7$ . We will break down each term and determine its coefficient and degree.

What are Coefficients and Degrees in Polynomials?

Coefficients: In a polynomial, a coefficient is a numerical value that is multiplied by a variable or a power of a variable. Coefficients can be positive or negative numbers.
Degrees: The degree of a term in a polynomial is the exponent of the variable or the power to which the variable is raised. The degree of a polynomial is the highest degree of any term in the polynomial.

Breaking Down the Polynomial

The given polynomial is $x^8 y - 3xy + 8x^4 - 7$ . Let's break down each term and identify its coefficient and degree.

Term 1: $x^8 y$

Coefficient: The coefficient of $x^8 y$ is 1, as there is no numerical value multiplied by the variable.
Degree: The degree of $x^8 y$ is 8 + 1 = 9, as the variable $x$ is raised to the power of 8 and multiplied by $y$ , which is a variable with a degree of 1.

Term 2: $-3xy$

Coefficient: The coefficient of $-3xy$ is -3, as the numerical value -3 is multiplied by the variable $x$ and $y$ .
Degree: The degree of $-3xy$ is 1 + 1 = 2, as both $x$ and $y$ are variables with a degree of 1.

Term 3: $8x^4$

Coefficient: The coefficient of $8x^4$ is 8, as the numerical value 8 is multiplied by the variable $x$ raised to the power of 4.
Degree: The degree of $8x^4$ is 4, as the variable $x$ is raised to the power of 4.

Term 4: $-7$

Coefficient: The coefficient of $-7$ is -7, as there is no variable multiplied by the numerical value -7.
Degree: The degree of $-7$ is 0, as there is no variable.

Conclusion

In conclusion, the coefficients and degrees of each term in the polynomial $x^8 y - 3xy + 8x^4 - 7$ are as follows:

Term	Coefficient	Degree
$x^8 y$	1	9
$-3xy$	-3	2
$8x^4$	8	4
$-7$	-7	0

Introduction

In our previous article, we discussed the concept of coefficients and degrees in polynomials. We broke down each term in the polynomial $x^8 y - 3xy + 8x^4 - 7$ and identified its coefficient and degree. In this article, we will provide a Q&A guide to help you better understand coefficients and degrees in polynomials.

Q: What is the coefficient of a term in a polynomial?

A: The coefficient of a term in a polynomial is a numerical value that is multiplied by a variable or a power of a variable. Coefficients can be positive or negative numbers.

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

A: To determine the degree of a term in a polynomial, you need to identify the exponent of the variable or the power to which the variable is raised. The degree of a polynomial is the highest degree of any term in the polynomial.

Q: What is the difference between the coefficient and degree of a term?

A: The coefficient of a term is a numerical value that is multiplied by a variable, while the degree of a term is the exponent of the variable or the power to which the variable is raised.

Q: Can a term have a coefficient of 0?

A: Yes, a term can have a coefficient of 0. For example, in the polynomial $x^2 + 0x + 3$ , the term $0x$ has a coefficient of 0.

Q: Can a term have a degree of 0?

A: Yes, a term can have a degree of 0. For example, in the polynomial $x^2 + 3x + 0$ , the term $0$ has a degree of 0.

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

A: To determine the degree of a polynomial, you need to identify the highest degree of any term in the polynomial. For example, in the polynomial $x^3 + 2x^2 + 3x + 1$ , the highest degree is 3, so the degree of the polynomial is 3.

Q: Can a polynomial have multiple terms with the same degree?

A: Yes, a polynomial can have multiple terms with the same degree. For example, in the polynomial $x^2 + 2x^2 + 3x + 1$ , both terms $x^2$ have the same degree of 2.

Q: How do I simplify a polynomial by combining like terms?

A: To simplify a polynomial by combining like terms, you need to identify terms with the same degree and coefficient, and then combine them by adding or subtracting their coefficients. For example, in the polynomial $x^2 + 2x^2 + 3x + 1$ , the terms $x^2$ can be combined to get $3x^2$ .

Conclusion

In conclusion, coefficients and degrees are fundamental concepts in polynomials that help us understand the structure and properties of polynomials. By identifying the coefficients and degrees of each term in a polynomial, we can better understand the polynomial and perform various mathematical operations, such as simplifying and solving equations.

Common Mistakes to Avoid

Confusing coefficients and degrees: Make sure to distinguish between coefficients and degrees, as they are two different concepts.
Not identifying like terms: Make sure to identify terms with the same degree and coefficient, and then combine them by adding or subtracting their coefficients.
Not simplifying polynomials: Make sure to simplify polynomials by combining like terms to get the simplest form of the polynomial.

Practice Problems

Identify the coefficient and degree of each term in the polynomial $x^3 + 2x^2 + 3x + 1$ .
Simplify the polynomial $x^2 + 2x^2 + 3x + 1$ by combining like terms.
Determine the degree of the polynomial $x^3 + 2x^2 + 3x + 1$ .
Identify the coefficient and degree of each term in the polynomial $x^2 + 3x + 0$ .
Simplify the polynomial $x^2 + 3x + 0$ by combining like terms.

Answer Key

Coefficient: 1, 2, 3, 1; Degree: 3, 2, 1, 0
$3x^2 + 3x + 1$
3
Coefficient: 1, 3, 0; Degree: 2, 1, 0
$x^2 + 3x$