Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, identify the degree of the polynomial. \(200 p-30 p^{2} m+40 m^{3}\)

Step-by-Step Solution

Answer

The degree of the polynomial is 3.

Step 1: Identify Each Term's Degree

The polynomial given is \(200p - 30p^2m + 40m^3\). To find the degree of each term, we sum the exponents of the variables in each term. For the first term \(200p\), the degree is 1 (since the exponent of \(p\) is 1). For the second term \(-30p^2m\), the degree is \(2 + 1 = 3\) (since the exponents of \(p\) is 2 and \(m\) is 1). For the third term \(40m^3\), the degree is 3 (since the exponent of \(m\) is 3).

Step 2: Determine the Polynomial Degree

The degree of the entire polynomial is determined by the term with the highest degree. In this case, the highest degree among the terms is 3, which comes from both the \(-30p^2m\) and \(40m^3\) terms.

Key Concepts

Degree of a Term

When working with polynomials, understanding the concept of a term's degree is essential because it informs us about the behavior and characteristics of the polynomial expression. Each term in a polynomial is a combination of a coefficient (a numerical value) and variables raised to exponents. The degree of a term is the sum of the exponents of all variables present in that term.
For example, in the term \(200p\), there is only one variable \(p\) with an exponent of 1. So, the degree of this term is 1. In

• For the term \(-30p^2m\), the exponents of \(p\) and \(m\) are 2 and 1, respectively. Therefore, we add these exponents together to get a degree of 3.
• Similarly, the term \(40m^3\) has an exponent of 3 for the variable \(m\), giving it a degree of 3.

Understanding the degree of a term helps in discovering the polynomial's overall degree.

Exponents in Polynomials

Exponents play a crucial role in determining the properties and functions of polynomials. At its core, an exponent indicates how many times a number or variable is multiplied by itself. In polynomial expressions, exponents are typically whole numbers and appear as superscripts attached to variables.
For example, in the polynomial \(200p - 30p^2m + 40m^3\), each variable is accompanied by an exponent:

• The variable \(p\) in the term \(200p\) is raised to the power of 1.
• In the term \(-30p^2m\), the variables \(p\) and \(m\) have exponents of 2 and 1, respectively.
• Finally, for the term \(40m^3\), the variable \(m\) is raised to the power of 3.

Understanding all exponents in a polynomial offers insights into the term's degree and ultimately the overall polynomial degree.

Identifying Polynomial Terms

A polynomial is composed of multiple terms. These terms are individual parts of the expression, separated by plus or minus signs. Identifying each term within a polynomial is the first step toward analyzing its structure and calculating the overall degree.
For example, consider the polynomial \(200p - 30p^2m + 40m^3\):

• The term \(200p\) is identified as the first term.
• \(-30p^2m\) is the second term, noting the negative sign as part of the term.
• \(40m^3\) is the third term, adding another individual piece to the polynomial.

Each identified term contributes to the overall expression and affects how we analyze the entire polynomial. Knowing how to identify these terms is crucial for understanding how to calculate degrees and perform further calculations on the polynomial.