Skill: Classify polynomial functions by degree

A polynomial function is classified by its degree, which is the greatest exponent of the variable with a nonzero coefficient after like terms are combined.

Degree	Classification
$0$	constant
$1$	linear
$2$	quadratic
$3$	cubic
$4$	quartic
$5$	quintic

Worked example

Classify the polynomial function

$$f(x)=3x^4-2x^3+5x^2-3x^4+7x^3-4x^2+6x-1.$$

Step 1: Combine like terms.

Group terms with the same exponent:

$$(3x^4-3x^4)+(-2x^3+7x^3)+(5x^2-4x^2)+6x-1$$

Simplify:

$$f(x)=5x^3+x^2+6x-1.$$

Step 2: Identify the greatest exponent with a nonzero coefficient.

The exponents are $3$, $2$, $1$, and $0$. The greatest exponent is $3$, so the degree is $3$.

Step 3: Name the function.

A degree-$3$ polynomial function is called a cubic function.

Therefore,

$$\boxed{f(x)\text{ is a cubic polynomial function.}}$$

Remember: classify a polynomial by its degree after simplifying, not by the number of terms or the largest exponent shown before like terms are combined.