# Real Zeros of a Polynomial Function

## Introduction

This section focus on factoring a polynomial as far as possible. We take advantage of a result from section 4.1 that says if f(x) is a polynomial and f(r) = 0 for a real number r, then x - r is a factor of f(x).

## Factor Theorem

The mentioned important fact is restated here as teh factor theorem:

Let f be a polynomial function. Then x - c is a factor of f(x) if and only if f(c) = 0.

## Theorems For Finding Zeros

The first theorem makes a statement about the number of zeros:

 Theorem A polynomial function of degree n cannot have more than n real zeros.


When a polynomial is written in standard form, decreasing degree order, we have even more information about the potential zeros of a polynomial.

 Theorem: Descartes' Rule of Signs
Let f denote a polynomial function in standard form.
The number of real positive zeros of f is equal to the number of variations of sign of the nonzero coefficients of f(x) or else equals that number
minus an even integer.



The number of negative zeros of f either equals the number of variations in the sign of the nonzero coefficients of f(-x) or else equals that number minus an even integer.

Example: Let ${\displaystyle f(x)=x^{3}-4x^{2}-6x+9}$

To count the number of positive roots, we note that there are 2 sign changes, from 1 to -4 and -6 to 9. So there either 2 or 0 positive roots.

To count the number of negative roots, we have to look at ${\displaystyle f(-x)=-x^{3}-4x^{2}+6x+9}$. Here we only have one sign change, from -4 to 6. So there is one negative root.

Finally we arrive at the most precise theorem for finding rational zeros.

 Rational Zeros Theorem
Let f be a polynomial function of degree 1 or higher of the form ${\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots +a_{1}x+a_{0}~a_{n}\neq 0~a_{0}\neq 0}$
where each coefficient is an integer. If ${\displaystyle {\frac {p}{q}}}$, in lowest terms, is a rational zero of f, then p must divide ${\displaystyle a_{0}}$,
and q must divide ${\displaystyle a_{n}}$