Mathematik-Online lexicon: Roots of Polynomials

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	Mathematik-Online lexicon:
	Roots of Polynomials

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Example 2:

$\displaystyle 4x^3-27x-27=0$

Multiply the equation by 2:

$\displaystyle 8x^3-54x-54=0$

Substitute:

$\displaystyle 2x=u$

$\displaystyle u^3-27u-54=0$

Possible integer roots:

$\displaystyle u=\pm1 \textnormal{, } \pm2 \textnormal{, } \pm3 \textnormal{, } \pm6 \textnormal{, } \pm9 \textnormal{, } \pm54$

$\displaystyle u=-3 : \qquad -27+71-54=0 \qquad \textnormal{true}$

$\displaystyle u_{1}=-3$

Polynomial division:

$\displaystyle (u^3-27u-54):(u+3)=u^2-3u-18$

By the quadratic formula:

$\displaystyle u^2-3u-18=0$

$\displaystyle u_{2}=-3 \qquad u_{3}=6$

Resubstitute

$\displaystyle x_{1,2}=-\frac{3}{2} \qquad x_{3}=3$

(Authors: Jahn/Knödler)

see also:

automatisch erstellt am 8. 7. 2004