Problem 6 - College Algebra with Corequisite Support

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: Identify Terms of the Polynomial

To determine the degree of a polynomial, we first identify all the individual terms present. In the given polynomial, the terms are: \(14m^{3}\), \(m^{2}\), \(-16m\), and \(8\).

Step 2: Determine the Degree of Each Term

For each term, identify the power of the variable \(m\). The term \(14m^{3}\) has a degree of 3, \(m^{2}\) has a degree of 2, \(-16m\) has a degree of 1, and the constant term \(8\) has a degree of 0.

Step 3: Identify the Highest Degree

The degree of a polynomial is the highest power of its variable. Among the degrees 3, 2, 1, and 0 from each term, the highest is 3.

Key Concepts

Polynomial Terms

A polynomial is an expression that consists of variables and coefficients, composed using addition, subtraction, multiplication, and non-negative integer exponents. In order to fully understand polynomials, it's important to break them down into their individual terms. Each term is made up of a coefficient and a variable raised to an exponent. Consider the polynomial:

• The example is given as: \(14m^{3} + m^{2} - 16m + 8\).
• Here, \(14m^{3}\) is one term, with 14 as the coefficient and 3 as the exponent of \(m\).
• The second term \(m^{2}\) has an implicit coefficient of 1 and an exponent of 2.
• The third term \(-16m\) includes \(-16\) as the coefficient with \(m\) raised to the power of 1.
• Lastly, the constant term 8 has no variable attached, which means it acts as a term with an exponent of 0.

Breaking down the polynomial into these parts allows us to analyze and find information such as the degree of each term, which is essential for the next steps in polynomial arithmetic.

Highest Degree

The degree of a polynomial indicates the highest power that the variable is raised to among all its terms. This is a critical characteristic because it helps define the polynomial's behavior and its shape when plotted as a function. Finding the degree involves looking at each term in the polynomial and identifying the exponent on the variable:

• For \(14m^{3}\), the degree is 3.
• For \(m^{2}\), the degree is 2.
• For \(-16m\), the degree is 1.
• Finally, for the constant \(8\), it has a degree of 0.

After identifying the degrees of all individual terms, the highest degree among them determines the degree of the entire polynomial. In our case, the highest degree is 3, which gives us the degree of the polynomial. This degree is crucial as it directly affects how a polynomial function behaves graphically and algebraically.

Polynomial Identification

Understanding polynomials involves recognizing different forms and identifying their characteristics, such as degree, number of terms, and leading coefficients. It's important to distinguish a polynomial not just by its appearance, but also by its terms and degree.

• The degree indicates the polynomial's highest power, which we've identified as 3 in this example.
• The leading term, in this case, is \(14m^{3}\), as it has the highest degree.
• Polynomials can be categorized based on the number of terms they possess. The example \(14m^{3} + m^{2} - 16m + 8\) is a polynomial with four terms, known as a "quartic polynomial."

Correctly identifying and understanding each of these elements helps in polynomial classification and simplifies various operations like addition, subtraction, multiplication, and factoring. By knowing how to identify and label these characteristics, students can more easily manage polynomial expressions and equations in algebra.