Problem 16 - 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

The expression is a polynomial with degree 5.

Step 1: Identify Terms of the Expression

Examine each term in the expression \( \pi x^{5}-\frac{1}{7} x+\sqrt{3} \). The terms are \( \pi x^5 \), \(-\frac{1}{7} x \), and \( \sqrt{3} \).

Step 2: Check If Expression is a Polynomial

A polynomial is an expression that consists of variables raised to non-negative integer powers and coefficients that are real numbers. The terms \( \pi x^5 \) and \(-\frac{1}{7} x \) fit this definition as they have integer exponents 5 and 1, respectively, and \( \pi \) and \(-\frac{1}{7} \) are real numbers. The constant \( \sqrt{3} \) is also a real number.

Step 3: Determine Degree of the Polynomial

The degree of a polynomial is the highest power of the variable in the expression. In the expression \( \pi x^5 - \frac{1}{7} x + \sqrt{3} \), the highest power of \( x \) is 5, hence the degree is 5.

Key Concepts

Degree of a Polynomial

Understanding the degree of a polynomial is crucial. It helps us determine the "size" or "growth rate" of polynomial expressions. Simply put, the degree of a polynomial is the highest power of the variable present in the polynomial terms.
Consider the expression from the exercise \( \pi x^{5} - \frac{1}{7}x + \sqrt{3} \). Here, the highest power of the variable \( x \) is 5, so the degree of this polynomial is 5. The degree is important because:

• It tells us the most significant term in the polynomial that dominates the behavior for large values of \( x \).
• A polynomial's degree gives insight into its basic shape and the number of roots it can have.
• It predicts the number of solutions (or roots) of the polynomial equation.

Hence, understanding the degree gives you a snapshot of what the polynomial can do.

Non-negative Integer Powers

Non-negative integer powers are what make a polynomial suitable to be called just that – a polynomial. All valid polynomial expressions must have variables raised to whole numbers starting from zero upwards.
This means you will see terms with powers like 0, 1, 2, or any other non-negative integer. In our exercise, the terms \( \pi x^5 \) and \(-\frac{1}{7} x \) both have non-negative integer powers of 5 and 1, respectively. This confirms their legitimacy as polynomial terms.

• Negative or fractional exponents are not allowed in polynomials.
• The significance lies in the fact that calculations and manipulations like differentiation or integration are simpler with integer powers.

By using non-negative integer powers, polynomials remain stable mathematical entities which are quite manageable and understandable.

Polynomial Terms

A polynomial is constructed from terms, each made up of a coefficient multiplied by the variable raised to a power. Understanding these terms is foundational for working with polynomials.
In the expression \( \pi x^{5} - \frac{1}{7}x + \sqrt{3} \), there are three distinct polynomial terms:

• \( \pi x^5 \): which has a coefficient of \( \pi \) and a power of 5.
• \(-\frac{1}{7} x \): a negative term with coefficient \(-\frac{1}{7} \) and a power of 1.
• \( \sqrt{3} \): which is a constant term without any variable.

Each term contributes to the overall behavior of the polynomial function.

• Terms with higher powers influence the polynomial more at extreme values of the variable.
• Constant terms define the shift or displacement when plotted on a graph.
• The nature of coefficients and powers determines the direction and symmetry of polynomial graphs.

Hence, scrutinizing these terms gives a clear understanding of the polynomial's characteristics.