PlanetMath: polynomial equation of odd degree

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polynomial equation of odd degree

(Theorem)

Theorem 1 The equation

$\displaystyle a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n = 0$

(1)

with odd degree

and real coefficients

( $a_0 \ne 0$ ) has at least one real root

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

$\displaystyle f(x) = a_0x^n[1+g(x)]$

where $\displaystyle g(x) := \frac{a_1}{x}\!+\cdots\!+\!\frac{a_{n-1}}{x^{n-1}}\!+\!\frac{a_n}{x^n}$ . But we have $\displaystyle\lim_{\vert x\vert\to\infty}g(x) = 0$ because

$\displaystyle \lim_{\vert x\vert\to\infty}\frac{a_i}{x^i} = 0$

for all $i = 1,\,...,\,n$ . Thus there exists an

such that

$\displaystyle \vert g(x)\vert < 1\,\,$ for $\displaystyle \,\, \vert x\vert \geqq M.$

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

sign $\displaystyle f(\pm M) = ($ sign $\displaystyle a_0)($ sign $\displaystyle (\pm M))^n\cdot 1 = ($ sign $\displaystyle a_0)(\pm 1)$

since

is odd. Therefore the real polynomial function

has opposite signs in the end points of the interval $[-M,\,M]$ . Thus the continuity of

guarantees, according to Bolzano's theorem, at least one zero

in that interval. So (1) has at least one real root

"polynomial equation of odd degree" is owned by pahio.

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Keywords:

odd degree, real coefficients

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Cross-references: Bolzano's theorem, interval, end points, opposite, polynomial function, left hand side, proof, root, coefficients, real, degree, odd, equation
There is 1 reference to this entry.

This is version 4 of polynomial equation of odd degree, born on 2006-02-04, modified 2006-09-22.
Object id is 7588, canonical name is PolynomialEquationOfOddDegree.
Accessed 2601 times total.

Classification:

AMS MSC:	12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )
	26A09 (Real functions :: Functions of one variable :: Elementary functions)
	26A15 (Real functions :: Functions of one variable :: Continuity and related questions )
	26C05 (Real functions :: Polynomials, rational functions :: Polynomials: analytic properties, etc.)

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