Problem 6 - 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. \(\begin{array}{ll}{\text { Polynomial }} & \quad{\text { Type Terms Degree }} \\\ { \sqrt{2} x-\sqrt{3}}\end{array}\)

Step-by-Step Solution

Answer

Binomial, Terms: \( \sqrt{2}x, -\sqrt{3} \), Degree: 1

Step 1: Determine the Type of Polynomial

First, identify the number of terms in the given polynomial \( \sqrt{2}x - \sqrt{3} \). A term is separated by a \(+\) or \(-\) sign. Here, there are two terms: \( \sqrt{2}x \) and \( -\sqrt{3} \). A polynomial with two terms is called a *binomial*.

Step 2: List the Terms

Next, list the terms of the polynomial. The terms are the components separated by the \(+\) or \(-\) signs, as identified in the previous step. For this polynomial, the terms are \( \sqrt{2}x \) and \( -\sqrt{3} \).

Step 3: Determine the Degree

The degree of a polynomial is determined by the term with the highest power of the variable. In \( \sqrt{2}x - \sqrt{3} \), the term \( \sqrt{2}x \) has the variable \( x \) raised to the power of 1. Therefore, the degree of this polynomial is 1.

Key Concepts

Binomial

A binomial is a type of polynomial that contains exactly two terms. Think of it as a duo of mathematical expressions joined together. They're typically separated by a plus or minus sign. In the polynomial world, binomials are quite common and useful, especially in expansion and factoring problems.

For example, in the polynomial \(\sqrt{2}x - \sqrt{3}\), we can clearly see two distinct terms: \(\sqrt{2}x\) and \(-\sqrt{3}\). The presence of these two terms makes it a binomial.

**Characteristics of a Binomial:**

• Consists of two terms.
• Terms are separated by a plus \((+)\) or minus \((-))\) sign.
• Each term can be a constant, a variable, or a combination of both.

Recognizing binomials is crucial, as they form the building blocks for more complex polynomials and are frequently used in algebraic manipulations.

Degree of Polynomial

The degree of a polynomial is a key concept that refers to the highest power of the variable present in the expression. It's like the rank or level of the polynomial. Knowing the degree tells us a lot about the polynomial's behavior and characteristics.

For example, in the polynomial \(\sqrt{2}x - \sqrt{3}\), the highest power of \(x\) is 1. Therefore, the degree of this polynomial is 1.

**Understanding Polynomial Degrees:**

• The degree is determined by the term with the highest power of the variable.
• For a term like \(x^2\), the power is 2, so the degree is 2.
• Constant terms (without variables) have a degree of 0.

Knowing the degree helps in understanding how the polynomial behaves, especially when graphing or solving equations.

Terms of Polynomial

In algebra, a polynomial is made up of terms which are the distinct parts separated by plus \((+)\) or minus \((-\)) signs. Each term can be a number, a variable, or a mix of both.

To illustrate, consider the polynomial \(\sqrt{2}x - \sqrt{3}\). Here, the polynomial consists of two terms: \(\sqrt{2}x\) and \(-\sqrt{3}\).

**Key Points about Polynomial Terms:**

• Terms are the building blocks of a polynomial.
• Each term can include coefficients (like \(\sqrt{2}\) or \(-\sqrt{3}\)).
• Variables in a term can have exponents, which influence the term's degree.

Understanding terms is important because they dictate the polynomial's degree and structure.