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

Verified step by step guidance

1. Step 1: Recall the definition of a polynomial function. A polynomial function is an expression of the form \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where the exponents are non-negative integers and the coefficients \( a_n, a_{n-1}, \ldots, a_0 \) are real numbers.
2. Step 2: Examine each term in the given function \( f(x) = x^{1/2} - 3x^2 + 5 \).
3. Step 3: Identify the exponents of each term: \( x^{1/2} \) has an exponent of \( 1/2 \), \( -3x^2 \) has an exponent of \( 2 \), and \( 5 \) is a constant term with an implicit exponent of \( 0 \).
4. Step 4: Determine if all exponents are non-negative integers. Notice that \( x^{1/2} \) has an exponent that is not an integer, which violates the condition for a polynomial function.
5. Step 5: Conclude that \( f(x) = x^{1/2} - 3x^2 + 5 \) is not a polynomial function because it contains a term with a non-integer exponent.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression that consists of variables raised to non-negative integer 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, a_(n-1), ..., a_0 are constants and n is a non-negative integer. Functions that include fractional or negative exponents do not qualify as polynomial functions.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial f(x) = 4x^3 + 2x^2 - x + 7, the degree is 3 because the highest exponent of x is 3. The degree provides important information about the behavior of the polynomial function, including the number of roots and the end behavior of the graph.

Identifying Polynomial Functions

To determine if a function is a polynomial, check for the presence of non-negative integer exponents and ensure that the function does not include variables in the denominator or under a radical. For instance, the function f(x) = x^(1/2) - 3x^2 + 5 contains a term with a fractional exponent (x^(1/2)), which disqualifies it from being a polynomial function. Thus, careful examination of each term is essential.