Polynomial function

Definition

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. It can be expressed in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ where $a_i$ are constants and $n$ is a non-negative integer.

5 Must Know Facts For Your Next Test

1. A polynomial function has the general form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$.
2. The degree of the polynomial is the highest power of the variable in the function.
3. Polynomial functions are continuous and smooth, meaning they have no breaks, holes, or sharp corners.
4. The end behavior of a polynomial function depends on the leading term; if $a_n > 0$ and $n$ is even, both ends rise to infinity.
5. The roots or zeros of a polynomial function are the values of $x$ for which $f(x) = 0$, and these can be real or complex numbers.

Review Questions

• What is the general form of a polynomial function?
• How do you determine the degree of a polynomial function?
• What does it mean for a polynomial function to be continuous and smooth?

Related terms

Degree of Polynomial: The highest power of the variable in the polynomial equation.

Leading Coefficient:

The coefficient of the term with the highest degree in a polynomial.

Root (or Zero): A value for which the polynomial evaluates to zero.