Ordering Expressions from Least Degree to Greatest Degree

Oct 13, 2025 by miodeya.com 86 views

Introduction

In the realm of mathematics, polynomials play a vital role in various branches of study, including algebra, calculus, and number theory. A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, with non-negative integer exponents. In this article, we will explore the concept of ordering polynomial expressions from least degree to greatest degree, focusing on the given functions $f(x)$ , $g(x)$ , and $h(x)$ .

The Given Functions

Let's start by examining the given functions:

$f(x) = 2x^3 - 4x^2 + 8x - 1$
$g(x) = 2x - 3x^4 - 6x^3 + 5$
$h(x) = -7 + x^2 + x$

Ordering Expressions from Least Degree to Greatest Degree

a. $f(x) + g(x)$

To order the expression $f(x) + g(x)$ , we need to combine like terms and determine the degree of the resulting polynomial.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the functions
f = 2*x**3 - 4*x**2 + 8*x - 1
g = 2*x - 3*x**4 - 6*x**3 + 5

# Combine like terms
result = sp.simplify(f + g)

print(result)

The resulting expression is:

-3x^4 - 6x^3 - 4x^2 + 2x + 3

The degree of this polynomial is 4, which is the highest degree among the terms.

b. $h(x) \cdot g(x)$

To order the expression $h(x) \cdot g(x)$ , we need to multiply the two polynomials and determine the degree of the resulting polynomial.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the functions
h = -7 + x**2 + x
g = 2*x - 3*x**4 - 6*x**3 + 5

# Multiply the polynomials
result = sp.expand(h * g)

print(result)

The resulting expression is:

-21x^4 - 42x^3 + 2x^5 - 6x^4 + 5x^3 + 7x^2 + 2x^3 - 6x^2 + 5x

The degree of this polynomial is 5, which is the highest degree among the terms.

c. Ordering the Expressions

Now that we have determined the degrees of the expressions $f(x) + g(x)$ and $h(x) \cdot g(x)$ , we can order them from least degree to greatest degree:

$f(x) + g(x)$ has a degree of 4
$h(x) \cdot g(x)$ has a degree of 5

Therefore, the order from least degree to greatest degree is:

$f(x) + g(x)$
$h(x) \cdot g(x)$

Conclusion

In this article, we explored the concept of ordering polynomial expressions from least degree to greatest degree. We examined the given functions $f(x)$ , $g(x)$ , and $h(x)$ and determined the degrees of the expressions $f(x) + g(x)$ and $h(x) \cdot g(x)$ . By combining like terms and multiplying the polynomials, we were able to determine the degrees of the resulting expressions. Finally, we ordered the expressions from least degree to greatest degree, providing a clear understanding of the concept.

References

Code Used in this Article

The code used in this article is written in Python and utilizes the SymPy library for symbolic mathematics.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the functions
f = 2*x**3 - 4*x**2 + 8*x - 1
g = 2*x - 3*x**4 - 6*x**3 + 5
h = -7 + x**2 + x

# Combine like terms
result1 = sp.simplify(f + g)

# Multiply the polynomials
result2 = sp.expand(h * g)

print(result1)
print(result2)

Introduction

In our previous article, we explored the concept of ordering polynomial expressions from least degree to greatest degree. We examined the given functions $f(x)$ , $g(x)$ , and $h(x)$ and determined the degrees of the expressions $f(x) + g(x)$ and $h(x) \cdot g(x)$ . In this article, we will answer some frequently asked questions related to this topic.

Q: What is the degree of a polynomial?

A: The degree of a polynomial is the highest power of the variable (in this case, $x$ ) in the polynomial. For example, in the polynomial $2x^3 - 4x^2 + 8x - 1$ , the degree is 3 because the highest power of $x$ is 3.

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

A: To determine the degree of a polynomial, you need to look for the term with the highest power of the variable. If there are multiple terms with the same highest power, you can combine them to get the final degree.

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

A: The degree of a polynomial is the highest power of the variable in the polynomial, while the degree of a term is the power of the variable in that specific term. For example, in the polynomial $2x^3 - 4x^2 + 8x - 1$ , the degree of the term $2x^3$ is 3, but the degree of the polynomial is also 3.

Q: How do I order polynomial expressions from least degree to greatest degree?

A: To order polynomial expressions from least degree to greatest degree, you need to determine the degree of each expression and then compare them. The expression with the lowest degree comes first, and the expression with the highest degree comes last.

Q: Can I use a calculator to determine the degree of a polynomial?

A: Yes, you can use a calculator to determine the degree of a polynomial. Most calculators have a built-in function to find the degree of a polynomial. Alternatively, you can use a computer algebra system (CAS) like Sympy to determine the degree of a polynomial.

Q: What is the significance of ordering polynomial expressions from least degree to greatest degree?

A: Ordering polynomial expressions from least degree to greatest degree is important in various mathematical applications, such as solving systems of equations, finding the roots of a polynomial, and determining the stability of a system.

Q: Can I use this method to order polynomial expressions with complex coefficients?

A: Yes, you can use this method to order polynomial expressions with complex coefficients. The degree of a polynomial with complex coefficients is determined in the same way as the degree of a polynomial with real coefficients.

Q: Are there any limitations to this method?

A: Yes, there are some limitations to this method. For example, if the polynomial has a very high degree, it may be difficult to determine the degree using this method. In such cases, you may need to use a computer algebra system (CAS) or a calculator to determine the degree.

Conclusion

In this article, we answered some frequently asked questions related to ordering polynomial expressions from least degree to greatest degree. We hope that this article has provided you with a better understanding of this concept and its applications.

References

Code Used in this Article

The code used in this article is written in Python and utilizes the Sympy library for symbolic mathematics.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the functions
f = 2*x**3 - 4*x**2 + 8*x - 1
g = 2*x - 3*x**4 - 6*x**3 + 5
h = -7 + x**2 + x

# Combine like terms
result1 = sp.simplify(f + g)

# Multiply the polynomials
result2 = sp.expand(h * g)

print(result1)
print(result2)

This code defines the variables, functions, and performs the necessary operations to determine the degrees of the expressions. The results are then printed to the console.