Determine which functions are polynomial functions. For those that are, identify the degree. $f\left(x\right)=5x^2+6x^3$

Verified step by step guidance

1. Recall that a polynomial function is a function of the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where each exponent is a non-negative integer and the coefficients \(a_i\) are real numbers.
2. Look at the given function: \(f(x) = 5x^2 + 6x^3\). Check the exponents of \(x\) in each term. Here, the exponents are 2 and 3, both of which are non-negative integers.
3. Since all terms have non-negative integer exponents and real coefficients, \(f(x)\) is a polynomial function.
4. To find the degree of the polynomial, identify the term with the highest exponent. In this case, the highest exponent is 3 from the term \$6x^3$.
5. Therefore, the degree of the polynomial function \(f(x) = 5x^2 + 6x^3\) is 3.

Key Concepts

Polynomial Functions

A polynomial function is a function that can be expressed as a sum of terms consisting of variables raised to non-negative integer powers multiplied by coefficients. For example, f(x) = 5x^2 + 6x^3 is a polynomial because the exponents are whole numbers and coefficients are real numbers.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the function with a non-zero coefficient. In f(x) = 5x^2 + 6x^3, the degree is 3 because the term 6x^3 has the highest exponent.

Identifying Polynomial Terms

To determine if a function is polynomial, check each term's exponent to ensure it is a non-negative integer and that the function does not include variables in denominators, roots, or negative exponents. Terms like x^2 and x^3 qualify, confirming the function is polynomial.