Problem 5 - College Algebra with Corequisite Support

Question

For the following exercises, identify the degree of the polynomial. \(7 x-2 x^{2}+13\)

Step-by-Step Solution

Answer

The degree of the polynomial is 2.

Step 1: Identify Each Term

The given polynomial is \(7x - 2x^2 + 13\). Identify each term: \(7x\), \(-2x^2\), and \(13\).

Step 2: Determine the Degree of Each Term

The degree of a term is determined by the exponent of the variable in the term. For \(7x\), the degree is 1. For \(-2x^2\), the degree is 2 because the exponent of \(x\) is 2. The term \(13\) is a constant, which means its degree is 0.

Step 3: Identify the Highest Degree

The degree of the polynomial is the highest degree among all the terms. Compare the degrees: 1 (from \(7x\)), 2 (from \(-2x^2\)), and 0 (from \(13\)). The highest degree is 2.

Key Concepts

Polynomial Terms

A polynomial is essentially a sum of various terms, each made up of variables and constants. Understandably, each term can have different structures.
It's crucial to identify each term to grasp the polynomial's behavior better. Let's break this down with an example:
In the polynomial \(7x - 2x^2 + 13\),

• Term 1: \(7x\) - This term features a variable \(x\) raised to the first power.
• Term 2: \(-2x^2\) - This is a polynomial term with \(x\) squared.
• Term 3: \(13\) - A constant term, lacking any variables.

Identifying and understanding each term gives us the insight needed to analyze the polynomial's overall structure and complexity.

Exponents

Exponents are vital for determining a term's degree within a polynomial. They indicate how many times a variable is multiplied by itself.
For example, in \(-2x^2\):

• The exponent is 2, telling us \(x\) is squared.

More generally, exponents guide us in understanding the behavior of the polynomial over different values.
In the terms:

• \(7x\) - The exponent of \(x\) is 1, indicating a linear relationship.
• \(-2x^2\) - The exponent is 2, suggesting a quadratic relationship.

Remember, if there's no exponent visible, as in \(13\), it implicitly means the variable (if present) is raised to the power of zero.
Exponents thus not only define the term's degree but help in plotting how a polynomial behaves when graphed.

Constant Term

A constant term in a polynomial is a term that contains no variables, only a fixed number. In essence, it acts as the y-intercept in the graph of a function.
Taking the polynomial \(7x - 2x^2 + 13\) as an example:

• The constant term is \(13\).

Since there are no variables involved, the constant term's degree is 0.
This means no matter the value of \(x\), the constant term always remains the same in contributing to the overall polynomial's value.
Its simple presence can shift the entire graph up or down when plotting the polynomial on a coordinate plane.
Understanding the constant is crucial for interpreting the full expression and its graph.