Problem 1 - College Algebra

Question

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=5 x^{2}+6 x^{3}$$

Step-by-Step Solution

Answer

The function \(f(x)=5 x^{2}+6 x^{3}\) is a polynomial function. The degree of the polynomial is 3.

Step 1: Compare the function to the polynomial function form

The general form of a polynomial function is \(p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{2} x^{2}+a_{1} x+a_{0}\) where \(a_{0}, a_{1},\ldots, a_{n}\) are constant coefficients and n is a nonnegative integer. When comparing the function \(f(x)=5 x^{2}+6 x^{3}\) to this form, we see the function complies with the general form of a polynomial function, it consists of finite amount of terms, and each term is made of a constant coefficient multiplied by a variable raised to a constant nonnegative integer power.

Step 2: Identify the degree of the polynomial

The degree of a polynomial is the power of the variable in the term of highest degree. The function \(f(x)=5 x^{2}+6 x^{3}\) has highest power as 3 in the term \(6 x^{3}\). So, the degree of this polynomial is 3.

Key Concepts

Identifying Polynomial Degree

Understanding the degree of a polynomial is essential as it conveys significant information about the function's behavior, particularly its growth rate and the number of roots it may possess. To identify the degree of a polynomial, we look for the term with the highest exponent of the variable. In the function f(x) = 5x^2 + 6x^3, the term with the highest exponent on x is 6x^3. The exponent here is 3, meaning the polynomial degree is 3. It's noteworthy that the degree tells us the polynomial will behave like x^3 for very large and very small values of x, indicating the function has a cubic nature.

As students delve into polynomial functions, grasping the significance of identifying the polynomial degree will aid in predicting the number of turning points on the graph, which can be at most one less than the degree of the polynomial. Hence, a third-degree polynomial could have up to two turns, shaping the graph's overall appearance.

Polynomial Function Form

A polynomial function can be recognized by its specific form, which must comply with certain rules. Typically, a polynomial is expressed as p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0, where a_n to a_0 are constants, and n is a nonnegative integer representing the polynomial's degree. Each term in the polynomial, known as a monomial, comprises a constant coefficient multiplied by the variable to a nonnegative integer power. This structure ensures the polynomial function is smooth and continuous over all real numbers.

When examining whether a given function is a polynomial, one should check for this general form. For instance, the function f(x) = 5x^2 + 6x^3 is indeed a polynomial since each term includes a variable raised to a nonnegative integer power and each term's coefficient is constant. Recognizing this structure is critical for students as they learn to differentiate polynomial functions from other types of functions.

Polynomial Coefficients

The coefficients of a polynomial are the numerical factors that multiply each term's variable. In the polynomial f(x) = 5x^2 + 6x^3, the coefficients are 5 and 6. These numbers are essential as they influence the shape and location of the function's graph. Coefficients can be positive or negative, and their magnitude affects how steep or flat the graph appears near the respective term's variable power.

Students should pay attention to coefficients since they play a crucial role in the polynomial's behavior. For example, if a coefficient is zero, the term does not appear in the function at all, reducing the polynomial's degree. On the contrary, a high coefficient value magnifies the impact of the respective term on the polynomial's graph. Recognizing and understanding these coefficients is a key skill when evaluating polynomials and predicting their graphs.