Properties of polynomial coefficients

A polynomial of degree $n$ can be written as $a_n x^n + a_{n-1} x^{n-1} + ext{...} + a_1 x + a_0$, where the $a_i$ are called the coefficients. The coefficients determine the behavior and properties of the polynomial.

Important properties

• The leading coefficient ($a_n$) affects the end behavior of the polynomial.

• The constant term ($a_0$) is the value of the polynomial at $x=0$.

• If the polynomial is factored as $a_n(x - r_1)(x - r_2) ext{...}(x - r_n)$, the coefficients relate to the roots $r_i$ through Vieta's formulas.

• Changing coefficients changes the graph and roots of the polynomial.