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:

TheoremA 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) pr else equals that number minus an even integer.


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

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 
$f(-x)=-x^{3}-4x^{2}+6x+9$. 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots   + a_1x + a_0 ~ a_n \neq 0 ~ a_0 \neq 0}

 where each coefficient is an integer. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  \frac{p}{q}}
, in lowest terms, is a rational zero of f, then p must divide Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0}
,
 and q must divide Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n}

 
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