MathAdmin at 04:07, 21 October 2015

2015-10-21T04:07:24Z

MathAdmin: Created page with "
TOC
==Introduction== This section focus on factoring a polynomial as far as possible. We take advantage of a result from section 4.1 that say..."

2015-10-21T04:04:21Z

Created page with "<div class="noautonum">__TOC__</div> ==Introduction== This section focus on factoring a polynomial as far as possible. We take advantage of a result from section 4.1 that say..."

New page

<div class="noautonum">__TOC__</div>
==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) pr else equals that number minus an even integer.

Example:
Let <math> f(x) = x^3 - 4x^2 - 6x + 9</math>

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
<math>f(-x) = -x^3 -4x^2 +6x + 9</math>. 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 <math>f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots
+ a_1x + a_0 ~ a_n \neq 0 ~ a_0 \neq 0</math>
where each coefficient is an integer. If <math> \frac{p}{q}</math>, in lowest terms, is a rational zero of f, then p must divide <math>a_0</math>,
and q must divide <math>a_n</math>

[[Math_5|'''Return to Topics Page]]

@@ Line 15: / Line 15: @@
 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.
+  '''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'''
+  '''Theorem: Descartes' Rule of Signs'''
-Let f denote a polynomial function in standard form.
+  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)
+  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
- or else equals that number minus an even integer.
+  minus an even integer.
- The number of negative zeros of f either equals the number of variations in the sign of the nonzero coefficients
+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:
+Example:
- Let <math> f(x) = x^3 - 4x^2 - 6x + 9</math>
+Let <math> f(x) = x^3 - 4x^2 - 6x + 9</math>
-−
- To count the number of positive roots, we note that there are 2 sign changes, from 1 to -4 and -6 to 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.
+So there either 2 or 0 positive roots.
- To count the number of negative roots, we have to look at
+To count the number of negative roots, we have to look at
- <math>f(-x) = -x^3 -4x^2 +6x + 9</math>. Here we only have one sign change, from -4 to 6. So there is one negative root.
+<math>f(-x) = -x^3 -4x^2 +6x + 9</math>. 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.
+Finally we arrive at the most precise theorem for finding rational zeros.
- '''Rational Zeros Theorem'''
+  '''Rational Zeros Theorem'''
- Let f be a polynomial function of degree 1 or higher of the form <math>f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots
+  Let f be a polynomial function of degree 1 or higher of the form <math>f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots
    + a_1x + a_0 ~ a_n \neq 0 ~ a_0 \neq 0</math>
    where each coefficient is an integer. If <math> \frac{p}{q}</math>, in lowest terms, is a rational zero of f, then p must divide <math>a_0</math>,
    and q must divide <math>a_n</math>
-  [[Math_5|'''Return to Topics Page]]
+[[Math_5|'''Return to Topics Page]]

Real Zeros of a Polynomial Function - Revision history

MathAdmin at 04:07, 21 October 2015

MathAdmin: Created page with "__TOC__ ==Introduction== This section focus on factoring a polynomial as far as possible. We take advantage of a result from section 4.1 that say..."

MathAdmin: Created page with "
TOC
==Introduction== This section focus on factoring a polynomial as far as possible. We take advantage of a result from section 4.1 that say..."