Polynomial Functions

Polynomial Function

Polynomial Function

A function expressed in the form \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \) with non-negative integer \( n \) and non-zero coefficient \( a_n \).

Degree

The highest power of \( x \) with a non-zero coefficient in a polynomial function.

Leading Coefficient

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

Root

A solution to the equation \( P(x) = 0 \) for a polynomial function.

Factor Theorem

If \( c \) is a root of \( P(x) \), then \( x - c \) is a factor of \( P(x) \).

End Behavior

Behavior of a polynomial function determined by the leading term \( a_nx^n \) for even/odd \( n \) and positive/negative \( a_n \).

Turning Points

Points where a polynomial function changes direction, with up to \( n-1 \) turning points for a polynomial of degree \( n \).

Intercepts

Points where a polynomial function crosses the x-axis (roots) and y-axis (constant term).