Problem 10 - College Algebra

Question

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=\frac{x^{2}+7}{3}$$

Step-by-Step Solution

Answer

The function \(f(x)=\frac{x^{2}+7}{3}\) is a polynomial function with degree 2.

Step 1: Determine the type of function

For the function to qualify as a polynomial function, it must satisfy the following conditions: 1. The exponents of the variable (x) need to be whole numbers (which includes zero). 2. The coefficients need to be real numbers. In the function \(f(x)=\frac{x^{2}+7}{3}\), the variable \(x\) is raised to the second power which is a whole number, and the coefficients 1 and 7 are real numbers.

Step 2: Check division

A common mistake is to think that if there is a division in the expression it is not a polynomial. However, a polynomial is allowed to have division, as long as the division is not with the variable \(x\). In the function \(f(x)=\frac{x^{2}+7}{3}\), only the whole expression is divided by a constant (3), but the variable \(x\) is not being divided. So, this division does not disqualify the function as a polynomial.

Step 3: Identify the degree

The degree of a polynomial function is determined by the highest exponent the variable occurs with. In the function \(f(x)=\frac{x^{2}+7}{3}\), the variable \(x\) has the highest exponent of 2, so the degree of this polynomial function is 2.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a key aspect that describes the polynomial itself. It is the highest power of the variable found in the entire expression. For example, in the polynomial \(f(x) = \frac{x^2 + 7}{3}\), we observe \(x\) raised to the power of 2. This 2 is the highest exponent present and hence, clarifies that the degree of the polynomial is 2.

Understanding the degree is crucial because it tells us several things about the polynomial:

• The number of solutions the polynomial might have (if real and distinct)
• The complexity of the graph representation, including how many times it may intersect the x-axis
• The end behavior, meaning how the graph behaves as \(x\) approaches positive or negative infinity

By recognizing the degree of a polynomial, we get insights into other characteristics like its slope at different points and possible maximum or minimum values, which allows for a comprehensive understanding of its graph.

Whole Number Exponents

While working with polynomials, it’s important to note that the exponents of the variables should always be whole numbers. Whole number exponents include 0, 1, 2, 3, and so forth.

This type of exponent ensures that we only have non-negative integer powers of the variable in the polynomial. This is essential because it affects how the polynomial graph looks and how it behaves. In the function \(f(x) = \frac{x^2 + 7}{3}\), the exponent of \(x\) is 2, which clearly fits the definition of a whole number.

Why is this important?

• It prevents division by zero, which would make the function undefined
• Ensures that no fractions or negative numbers are used as exponents, which would then transform a polynomial into some other type of function

Keeping exponents as whole numbers keeps the function well-behaved and predictable, which is crucial for correctly analyzing polynomial expressions.

Real Number Coefficients

In any polynomial, the coefficients should always be real numbers. Real numbers include all numbers on the number line, incorporating both positive and negative numbers, as well as fractions and decimals. This broad category excludes imaginary numbers such as \(i\).

In the expression \(f(x) = \frac{x^2 + 7}{3}\), the coefficients 1 and 7 (before dividing by 3) are real numbers, confirming the polynomial nature of \(f(x)\).

Why is it important for coefficients to be real numbers?

• It keeps the polynomial linear in terms of calculations with real values
• Facilitates straightforward graphing on a traditional x-y (Cartesian) plane
• Assists in predicting the roots and other intersection points of the polynomial

For effective problem solving, ensuring your coefficients are real numbers leads to more manageable calculations and simplifies interpreting the behavior of polynomial functions.