Problem 6 - College Algebra

Question

For the following exercises, identify the degree of the polynomial. $$ 14 m^{3}+m^{2}-16 m+8 $$

Step-by-Step Solution

Answer

The degree of the polynomial is 3.

Step 1: Understanding the Polynomial

The given polynomial is \(14m^3 + m^2 - 16m + 8\). A polynomial is an expression made up of terms where each term is a product of a number (called the coefficient) and one or more variables raised to a whole number power.

Step 2: Identifying the Degree of Each Term

Each term in the polynomial has a degree given by the exponent of the variable. For the term \(14m^3\), the degree is 3. For the term \(m^2\), the degree is 2. For the term \(-16m\), the degree is 1. For the constant term \(8\), which can be considered as \(8m^0\), the degree is 0.

Step 3: Determine the Degree of the Polynomial

The degree of a polynomial is defined as the highest degree of any term in the polynomial. Look at the degrees we found: 3, 2, 1, and 0. The highest degree among these is 3.

Key Concepts

Polynomial Terms

A polynomial is made up of multiple terms. Each term is a part of the expression that consists of a coefficient and variables raised to an exponent.
For the polynomial expression \(14m^3 + m^2 - 16m + 8\), it's important to recognize its four separate terms:

• \(14m^3\): In this term, 14 is the coefficient and \(m\) is the variable raised to the 3rd power.
• \(m^2\): Here, the coefficient is 1 (which is often not written), with \(m\) raised to the power of 2.
• \(-16m\): The coefficient is -16, and the variable \(m\) is raised to the power of 1.
• 8: This is a constant term because it lacks a variable. It can be thought of as \(8m^0\), since anything raised to the power of 0 is 1.

Recognizing each part helps break down the expression for better understanding.

Polynomial Degree

The degree of a polynomial is a critical feature that tells us about the behavior of the polynomial across its terms. The degree of a term within a polynomial is determined by the exponent of its variable.
The degree in a polynomial shows the highest power of the variable represented in any of its terms:

• In the term \(14m^3\), the degree is 3 because the variable \(m\) is raised to the power of 3.
• For \(m^2\), the degree is 2.
• The term \(-16m\) has a degree of 1, with the variable \(m\) at the power of 1.
• Lastly, the constant term \(8\) is at degree 0, represented as \(8m^0\).

Understanding the degree helps in assessing the polynomial's influence, especially in terms of its shape and the speed at which it grows or shrinks.

Identifying Polynomial Degree

Identifying the complete degree of the polynomial itself is straightforward once the degrees of the individual terms are clarified.
The degree of the entire polynomial is simply the highest degree of any of its individual terms. Put simply, it's about seeking the largest exponent:

• From our breakdown, the degrees are 3, 2, 1, and 0.
• The highest of these is 3, originating from the term \(14m^3\).

Hence, the degree of the polynomial \(14m^3 + m^2 - 16m + 8\) is 3.
Knowing the highest degree gives insight into the polynomial's leading behavior and how it might behave as the variable value becomes very large or very small.