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

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Step 1: Recall the definition of a polynomial function. A polynomial function is an expression of the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are constants and \( n \) is a non-negative integer.

Step 2: Examine the given function \( f(x) = 5x^2 + 6x^3 \). Identify the terms and their respective powers of \( x \).

Step 3: Check if all the exponents of \( x \) are non-negative integers. In this function, the exponents are 2 and 3, which are non-negative integers.

Step 4: Since all the exponents are non-negative integers, \( f(x) = 5x^2 + 6x^3 \) is a polynomial function.

Step 5: Identify the degree of the polynomial. The degree is the highest power of \( x \) in the function, which is 3 in this case.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Functions

A polynomial function is a mathematical expression that involves variables raised to whole number powers, combined using addition, subtraction, and multiplication. The general form is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n are coefficients and n is a non-negative integer. Polynomial functions are continuous and smooth, and they can be classified by their degree.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial f(x) = 5x^2 + 6x^3, the term with the highest exponent is 6x^3, making the degree of this polynomial 3. The degree provides important information about the polynomial's behavior, including the number of roots and the end behavior of the graph.

Identifying Polynomial Functions

To determine if a function is a polynomial, check that all terms are in the form of a_n*x^n, where n is a non-negative integer. Functions that include variables in the denominator, negative exponents, or fractional exponents are not polynomials. In the given function f(x) = 5x^2 + 6x^3, both terms meet the criteria, confirming it as a polynomial function.

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