Problem 37 - Elementary and Intermediate Algebra

Question

Find the degree of each polynomial. See Example \(1 .\) $$ \frac{1}{3} x-5 $$

Step-by-Step Solution

Answer

The degree of the polynomial \( \frac{1}{3}x - 5 \) is 1.

Step 1: Identify the Terms

Look at the polynomial \( \frac{1}{3}x - 5 \). This polynomial consists of two terms: \( \frac{1}{3}x \) and \(-5 \).

Step 2: Determine the Degree of Each Term

For the term \( \frac{1}{3}x \), the variable \(x\) is raised to the power of 1 (since \( x = x^1 \)), so it has a degree of 1. The constant term \(-5\) has no variable attached to it, so it has a degree of 0.

Step 3: Find the Highest Degree

Among all terms, identify the term with the highest degree. Here, \( \frac{1}{3}x \) has the highest degree of 1. Therefore, the degree of the entire polynomial is the highest degree among its terms.

Key Concepts

Understanding Polynomials

Polynomials are mathematical expressions consisting of variables, coefficients, and arithmetic operations such as addition, subtraction, and multiplication. They are composed of one or more terms. Each term is a product of a constant coefficient and a variable raised to a non-negative integer exponent. Polynomials can have multiple terms, and they vary in complexity, from simple ones like linear polynomials to more complex forms like quadratics and cubics.

Key properties of polynomials include:

• Each term is separated by either a plus or a minus sign.
• The exponents of the variables are always non-negative integers.
• Polynomials cannot have variables in the denominator or involve negative exponents.

Recognizing these properties will help you identify and work with polynomials effectively.

Understanding the Degree of a Term

The degree of a term in a polynomial is defined by the exponent of its variable. For single-variable terms like \( a \cdot x^n \), the degree is simply the power \( n \). The degree of a term tells us how the term will behave as the variable changes, especially at larger values.

Here's how to find the degree of a term:

• If a term is \( c \cdot x^1 \), its degree is 1 because the variable \( x \) is raised to the first power.
• If a term contains a constant like \( b \), its degree is zero because it essentially represents \( b \cdot x^0 \).
• Terms with higher exponents will dominate the behavior of a polynomial as the variable gets larger.

To determine the degree of a polynomial, identify the term with the highest degree and that defines the degree of the polynomial as a whole.

What is the Constant Term?

A constant term in a polynomial is a term that has no variable attached to it. It is significant because it represents a fixed number regardless of the value of the variable. In a polynomial expression, it looks like a standalone number. For instance, in the polynomial \( \frac{1}{3}x - 5 \), "-5" is the constant term.

Important aspects to understand about constant terms:

• They have a degree of 0 because they can be thought of as being multiplied by \( x^0 \).
• Constant terms act as vertical shifts in the graph of a polynomial function. Changing the constant term will move the graph up or down but will not affect its shape or direction.
• In real-world scenarios, constant terms can represent starting values or baselines.

Recognizing and understanding constant terms is vital in simplifying and solving polynomial equations.