In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. g(x)=6x⁷+πx⁵+2/3 x

Verified step by step guidance

1. Recall that a polynomial function is a function of the form $f\left(x\right)=a_n{x}^n$, where the exponents $n$ are whole numbers (non-negative integers) and the coefficients $a$ are real numbers.
2. Examine the given function . Check each term to see if the exponents of $x$ are whole numbers and the coefficients are real numbers.
3. Identify the exponents of $x$ in each term: $7$ in $6x^7$, $5$ in , and $1$ in . All are whole numbers.
4. Confirm that the coefficients $6$, (pi), and are real numbers, which they are.
5. Since all terms meet the criteria for a polynomial, conclude that $g(x)$ is a polynomial function. The degree of the polynomial is the highest exponent of $x$, which is $7$.

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. The coefficients can be any real numbers, including constants like π or fractions. For example, g(x) = 6x^7 + πx^5 + (2/3)x is a polynomial because all 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 expression with a non-zero coefficient. It indicates the polynomial's order and affects its graph's shape and behavior. In g(x) = 6x^7 + πx^5 + (2/3)x, the degree is 7, since the highest exponent of x is 7.

Identifying Non-Polynomial Functions

To determine if a function is not a polynomial, check for variables with negative, fractional, or variable exponents, or variables in denominators, roots, or absolute values. If any such terms exist, the function is not polynomial. Since g(x) has only positive integer exponents and no variables in denominators, it qualifies as a polynomial.