Problem 18 - 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 of degree 1.

Step 1: Identify the components of the expression

Examine the given expression: \(\frac{1}{\sqrt{7}} x + \frac{1}{2}\). This expression consists of two terms. The first term is \(\frac{1}{\sqrt{7}} x\), and the second term is \(\frac{1}{2}\).

Step 2: Determine the terms of the expression

A polynomial is defined as an expression consisting of variables and coefficients, involving only non-negative integer powers of the variables. Check each term:- The term \(\frac{1}{\sqrt{7}} x\) has the variable \(x\) raised to the power of 1 (a non-negative integer).- The term \(\frac{1}{2}\) is a constant, which is acceptable in a polynomial.

Step 3: Identify the coefficients

For a term to be part of a polynomial, its coefficient can be any real number. Here:- \(\frac{1}{\sqrt{7}}\) is the coefficient of \(x\). This is a real number.- \(\frac{1}{2}\) is a constant term, which is also a real number.

Step 4: Check if expression is a polynomial

The expression \(\frac{1}{\sqrt{7}} x + \frac{1}{2}\) is in the form of a polynomial because it meets the criteria: it has terms with non-negative integer powers of \(x\) and constant coefficients that are real numbers.

Step 5: Determine the degree of the polynomial

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

Key Concepts

degree of a polynomial

The degree of a polynomial is a key concept in understanding polynomial expressions. It refers to the highest power of the variable in a polynomial. In simpler terms, look for the largest exponent attached to the variable.
For example, in the expression \(3x^2 + 2x + 1\), the term with the highest power of \(x\) is \(3x^2\). Here, the degree is 2 because the exponent of \(x\) in this term is 2.
When checking the degree, make sure:

• The variable's power must be a non-negative integer.
• Constant terms like \(7\) or \( -3\), which can be seen as \(7x^0\) or \(-3x^0\), have degrees of 0.
  

Determining the degree isn't just a formality; it tells us about the polynomial's behavior and its graph's shape. For instance, polynomials of degree 1 are straight lines, while those of degree 2 are parabolas.

real numbers

In mathematics, real numbers are numbers that can be found on the number line. They include both rational numbers, such as fractions and integers, and irrational numbers, which cannot be expressed as simple fractions.
Some characteristics of real numbers include:

• Real numbers can be positive, negative, or zero.
• They can be written in decimal form, either terminating like 0.75 or repeating like 0.333...
• Common examples include integers like -9, rational numbers like \(\frac{1}{2}\), and irrational numbers like \(\sqrt{2}\).
  

Real numbers are crucial when discussing polynomials, as they often comprise the coefficients and constant terms. Understanding real numbers helps us analyze polynomial expressions correctly, knowing that they can handle a broad range of values.

non-negative integer powers

Non-negative integer powers are a fundamental part of polynomials. They mean that the exponent used on a variable is either zero or a positive whole number.
For example, \(x^0, x^1, x^2\) are all non-negative integer powers because their exponents are 0, 1, and 2 respectively.
Why is this important?

• Polynomials require their variables to have non-negative integer powers. For instance, \(x^{-1}\) is not allowed because \(-1\) is a negative exponent.
• Non-negative integer powers ensure the polynomial expression remains defined and operates smoothly for any real number inputs.
  

When analyzing polynomial functions, always ensure each term's power adheres to this rule, as it defines the "polynomial" nature of the expression.