Problem 10 - Algebra and Trigonometry

Question

Determine whether the expression is a polynomial. If it is, state its degree. $$ \pi x^{5}-\frac{1}{7} x+\sqrt{3} $$

Step-by-Step Solution

Answer

Yes, it is a polynomial of degree 5.

Step 1: Identify the Terms

Break down the expression into separate terms: \( \pi x^5 \), \( -\frac{1}{7}x \), and \( \sqrt{3} \). These terms are separated by addition and subtraction.

Step 2: Analyze Each Term

Examine each term to determine if it fits the form of a polynomial term, which is \( ax^n \) where \( a \) is a constant and \( n \) is a non-negative integer:1. \( \pi x^5 \): This is acceptable as a polynomial term (coefficient \( \pi \) and exponent 5).2. \( -\frac{1}{7}x \): This is also a polynomial term (coefficient \( -\frac{1}{7} \) and exponent 1).3. \( \sqrt{3} \): This is a constant term, which is acceptable in a polynomial expression.

Step 3: Verify Polynomial Criteria

Check to ensure all terms comply: All exponents of the variable \( x \) in its terms are non-negative integers (5 and 1), and \( \sqrt{3} \) is a constant (\( x^0 \)). Therefore, the expression is a polynomial.

Step 4: Determine the Degree of the Polynomial

Identify the degree of the polynomial by looking for the highest power of \( x \) in the expression. Among \( \pi x^5 \), \(-\frac{1}{7}x\), and \(\sqrt{3}\), the highest power is 5, so the degree of the polynomial is 5.

Key Concepts

Degree of Polynomial

In any polynomial, the degree is the highest power of the variable present in the expression. Understanding this is crucial, as it gives insight into the behavior and properties of polynomials. For instance, in the expression \( \pi x^5 - \frac{1}{7}x + \sqrt{3} \), we identify each term separately:

• \( \pi x^5 \)
• \(-\frac{1}{7}x\)
• \(\sqrt{3}\)

The degree is determined by the term with the highest power of \( x \), which is \( \pi x^5 \) with a power of 5.

Therefore, the degree of this polynomial is 5. This means that the polynomial will most prominently feature behavior corresponding to \( x^5 \), influencing its shape and interaction with the x-axis.

Non-Negative Integer Exponents

A polynomial requires that each term's exponent of the variable be a non-negative integer. This means that the exponents must be whole numbers like 0, 1, 2, etc., and cannot be negative or fractions.

In the expression \( \pi x^5 - \frac{1}{7}x + \sqrt{3} \), we see:

• \( x^5 \) with exponent 5
• \( x^1 \) (implied) with exponent 1

Both exponents are non-negative integers, which confirms these terms meet the criteria for a polynomial.

These conditions ensure that for any polynomial, you can effectively apply arithmetic operations and predict its behavior continuously without resulting in irregular or undefined values.

Constant Polynomial Term

Polynomials can also include terms without a variable, known as constant terms. These essentially represent coefficients multiplied by \( x^0 \), which equals 1.

In the example \( \pi x^5 - \frac{1}{7}x + \sqrt{3} \), the constant term is \( \sqrt{3} \). This is valid in a polynomial because it can be seen as:

• \( \sqrt{3} \times x^0 \)

The inclusion of constant terms in polynomials allows them to represent realistic scenarios, such as fixed values in a simulation or baseline levels in an experiment. Constant terms also modify the vertical position of the polynomial graph on a coordinate plane, playing a pivotal role in its interpretative significance.