Problem 12 - Algebra and Trigonometry

Question

Determine whether the expression is a polynomial. If it is, state its degree. $$ \frac{1}{\sqrt{7}} x+\frac{1}{2} $$

Step-by-Step Solution

Answer

The expression is a polynomial with a degree of 1.

Step 1: Understanding Polynomials

A polynomial is an algebraic expression that consists of terms which are either constants or variables raised to non-negative integer powers. The standard form is \( a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \).

Step 2: Identify Terms in the Expression

The given expression is \( \frac{1}{\sqrt{7}} x + \frac{1}{2} \). There are two terms: \( \frac{1}{\sqrt{7}} x \) and \( \frac{1}{2} \).

Step 3: Check the Terms for Polynomial Criteria

Both terms need to be checked to see if they fit the criteria of a polynomial. The term \( \frac{1}{\sqrt{7}} x \) has a variable term \( x \) raised to the power of 1, which is a non-negative integer, hence it satisfies polynomial criteria. The term \( \frac{1}{2} \) is a constant term, which is acceptable in polynomials.

Step 4: Calculate the Degree of the Polynomial

The degree of a polynomial is the highest power of the variable in the expression. In the term \( \frac{1}{\sqrt{7}} x \), the variable \( x \) is raised to the power of 1. Thus, the degree of the polynomial is 1.

Key Concepts

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and mathematical operations like addition and multiplication. Expressions can stand alone without an equals sign for any equation or inequality. For example, in the expression \[ \frac{1}{\sqrt{7}} x + \frac{1}{2} \], we see two things happening:

• A constant, \( \frac{1}{2} \), which does not change.
• A variable term, \( \frac{1}{\sqrt{7}} x \), where \( x \) can change based on different values we give it.

An expression becomes much more versatile by having these components as it can describe various relationships depending on the variable's input value. When discussing polynomials, understanding an algebraic expression is essential because polynomials are a type of algebraic expression.

Degree of a Polynomial

The degree of a polynomial points to the highest power of the variable in any given term. It's essentially a simple numeric value that tells us how the expression behaves at larger scales. For our example, \[ \frac{1}{\sqrt{7}} x + \frac{1}{2} \], the highest power of \( x \) is 1, found in the term \( \frac{1}{\sqrt{7}} x \).

• This means the polynomial's degree is 1.
• When determining the behavior or complexity of an expression, the degree provides key insights.

A higher degree usually indicates a more complex relationship or varying dynamic between variables and constants in the polynomial. In our case, since the degree is 1, we call this a linear polynomial, which implies it will produce a straight-line graph when plotted.

Non-Negative Integer Powers

Understanding the concept of non-negative integer powers is vital in identifying polynomials. Non-negative integer powers refer to exponents that are whole numbers starting from zero and moving upwards (e.g., 0, 1, 2, 3, etc.).

• This specification excludes negative exponents and fractions in the exponents.
• These powers help maintain the polynomial's continuity and smoothness when represented graphically.

In the expression \[ \frac{1}{\sqrt{7}} x + \frac{1}{2} \], the power of \( x \) is 1, a non-negative integer. This adheres to the basic rule for a polynomial that prohibits exponents like \( -1 \) or \( \frac{1}{2} \), ensuring both the expression and its graph share simplicity and predictability.