Problem 2 - College Algebra

Question

Determine which functions are polynomial functions. For those that are, identify the degree. $$ f(x)=7 x^{2}+9 x^{4} $$

Step-by-Step Solution

Answer

The function \( f(x) = 7x^{2} + 9x^{4} \) is a polynomial function. The degree of the polynomial is 4.

Step 1: Identifying Polynomial Function

The function provided is \( f(x) = 7x^{2} + 9x^{4} \). This function meets the definition of a polynomial as it is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Therefore, \( f(x) = 7x^{2} + 9x^{4} \) is a polynomial function.

Step 2: Determine the Degree

The degree of a polynomial function is the highest power of the variable \( x \) in the polynomial. In this case, the highest power of \( x \) in the polynomial \( f(x) = 7x^{2} + 9x^{4} \) is 4. Therefore, the degree of the polynomial is 4.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a fundamental concept in understanding polynomial functions. The degree is the highest power of the variable in the polynomial. It's important because it tells us about the polynomial's behavior and how many roots or solutions it might have. In an expression like \( f(x) = 7x^{2} + 9x^{4} \), to find the degree, we simply look for the term with the highest exponent.
In this case, we see that the term \( 9x^4 \) has the highest power, which is 4. Hence, the degree of the polynomial \( f(x) \) is 4. This tells us that the graph of this function will exhibit certain behaviors typical for a fourth-degree polynomial.

Non-negative Integer Exponents

Non-negative integer exponents are essential when working with polynomial functions. They are exponents that are either zero or positive whole numbers. This is crucial because, in a polynomial function, the variables can only have these types of exponents.
For instance, looking at \( f(x) = 7x^{2} + 9x^{4} \), both exponents, 2 and 4, are examples of non-negative integers. This is a defining feature of polynomial expressions: any variable raised to an exponent, such as \( x^2 \) or \( x^4 \), must use whole number exponents. Negative or fraction exponents would render the function a non-polynomial.

Variables and Coefficients

Understanding variables and coefficients is key to grasping polynomial functions. A variable is a symbol, usually \( x \), that represents a number in a polynomial. Coefficients are the numerical factors that multiply the variables.
In the expression \( f(x) = 7x^{2} + 9x^{4} \), we have two terms. The variable in both terms is \( x \). The coefficients are 7 and 9, respectively. The coefficient indicates how many times the variable's term is counted.
Thus, the term \( 7x^2 \) means 7 times \( x^2 \), and \( 9x^4 \) means 9 times \( x^4 \). In polynomial expressions, each term's structure follows this "coefficient times variable raised to a power" format.