An -th-degree polynomial in the variable is a function of the form
where the coefficients , , , are real or complex numbers with .
The zeros of a function are those -values for which . The real zeros of correspond to the -intercepts of the graph of : if the number is a zero of , then is an -intercept of the graph.
To find the zeros of a polynomial function, we must solve a polynomial equation of the form
General polynomial equations are often solved by first making one side of the equation equal to zero. After doing so, the next steps may vary depending on the nature of the polynomial. For now, we will focus on solving by factoring.
When a polynomial is completely factored, the number of times a specific linear factor occurs in the factorization is called the multiplicity of the corresponding zero.
The general shape of the graph of a polynomial can be easily determined from the polynomial's factored form.
The graph crosses the -axis at every zero of multiplicity 1.
The graph flattens and crosses the -axis at every zero of odd multiplicity.
The graph flattens, touches, and bounces off the -axis at every zero of even multiplicity.
The end behavior (the behavior as ) of the graph of is identical to that of , where is the degree and is the leading coefficient: