Problem 12 - Algebra and Trigonometry

Question

Complete the following table by stating whether the polynomial is a monomial, binomial, or trinomial, then list its terms and state its degree. \(\sqrt{2} x-\sqrt{3}\)

Step-by-Step Solution

Answer

It's a binomial with terms \( \sqrt{2}x \) and \( -\sqrt{3} \), of degree 1.

Step 1: Identify the Type of Polynomial

The expression given is \( \sqrt{2}x - \sqrt{3} \). A polynomial is categorized based on the number of terms it has. This expression consists of two terms: \( \sqrt{2}x \) and \( -\sqrt{3} \). An expression with two terms is called a **binomial**.

Step 2: List the Terms

In the expression \( \sqrt{2}x - \sqrt{3} \), the terms are the distinct elements separated by a plus or minus sign. Thus, the terms of the polynomial are \( \sqrt{2}x \) and \( -\sqrt{3} \).

Step 3: Determine the Degree of the Polynomial

The degree of a polynomial is the highest degree of its terms when simplified. Each term's degree is determined by the sum of the powers of its variables. In \( \sqrt{2}x \), the degree is 1. In \( -\sqrt{3} \), there is no variable \((x^0)\), so the degree is 0. The highest degree among the terms is thus 1. Therefore, the polynomial's degree is **1**.

Key Concepts

Binomial

A binomial is a type of polynomial with exactly two terms. These terms are separated by either a plus or minus sign. For instance, the expression \( \sqrt{2}x - \sqrt{3} \) is a binomial because it has two distinct terms. Binomials are a special case within polynomials, known for their simplicity and fundamental role in algebra.

• Binomials are the simplest multi-term polynomials.
• Common examples include expressions like \( x + y \) and \( 3a - 2b \).
• Understanding binomials is crucial for tackling more complex polynomial operations.

Recognizing binomials is important because many algebraic techniques, such as factoring and expanding, start with these basic forms.

Polynomial Terms

Terms in a polynomial are the separate components that are added or subtracted to form the complete expression. They usually involve coefficients and variables raised to some power.
In the expression \( \sqrt{2}x - \sqrt{3} \), the terms are:

• \( \sqrt{2}x \)
• \( -\sqrt{3} \)

Each term plays a crucial role in the structure and the properties of the polynomial. Terms tell us about the makeup of the polynomial and are crucial for finding sums, differences, products, and, importantly, degrees. The presence of different terms also helps in identifying the type of polynomial.

Degree of Polynomial

The degree of a polynomial is a measure of the "highest power" of its variable(s). This concept is fundamental as it tells us about the polynomial's behavior and its graph's shape.
To determine the polynomial's degree, inspect the terms to identify the highest power of the variable present. Consider \( \sqrt{2}x - \sqrt{3} \):

• The term \( \sqrt{2}x \) involves a variable raised to the first power, making its degree 1.
• The term \( -\sqrt{3} \) contains no variables \((x^0)\), thus its degree is 0.

The highest degree among these terms is 1. Therefore, the degree of the polynomial is 1. Understanding the polynomial's degree is crucial for insight into its potential roots and the general appearance of its graph.