## You are here

Homepolynomial equation of odd degree

## Primary tabs

# polynomial equation of odd degree

###### Theorem.

Proof. Denote by $f(x)$ the left hand side of (1). We can write

$f(x)=a_{0}x^{n}[1+g(x)]$ |

where $\displaystyle g(x):=\frac{a_{1}}{x}\!+\cdots\!+\!\frac{a_{{n-1}}}{x^{{n-1}}}\!% +\!\frac{a_{n}}{x^{n}}$. But we have $\displaystyle\lim_{{|x|\to\infty}}g(x)=0$ because

$\lim_{{|x|\to\infty}}\frac{a_{i}}{x^{i}}=0$ |

for all $i=1,\,...,\,n$. Thus there exists an $M>0$ such that

$|g(x)|<1\,\,\mbox{for}\,\,|x|\geqq M.$ |

Accordingly $1+g(\pm M)>0$ and

$\mbox{sign}f(\pm M)=(\mbox{sign}a_{0})(\mbox{sign}(\pm M))^{n}\cdot 1=(\mbox{% sign}a_{0})(\pm 1)$ |

since $n$ is odd. Therefore the real polynomial function $f$ has opposite signs in the end points of the interval $[-M,\,M]$. Thus the continuity of $f$ guarantees, according to Bolzano’s theorem, at least one zero $x$ of $f$ in that interval. So (1) has at least one real root $x$.

Keywords:

odd degree, real coefficients

Related:

AlgebraicEquation, ExampleOfSolvingACubicEquation

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

26A15*no label found*26A09

*no label found*12D10

*no label found*26C05

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Oct 21

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag