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Minggu, 21 Oktober 2012

Teorema Vieta

Misalkan$latex P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_{1}x+a_{0}$ adalah sukubanyak (polinom) dengan akar-akar $latex x_{1},\ x_{2},\ x_{3},\ ...,\ x_{n-1},\ x_{n}$
maka berlaku:
$latex x_{1}+x_{2}+x_{3}+...+x_{n-1}+x_{n}=-\frac{a_{n-1}}{a_{n}}$
$latex x_{1}x_{2}+x_{1}x_{3}+...+x_{2}x_{3}+x_{2}x_{4}+...+x_{n-1}x_{n}=+\frac{a_{n-2}}{a_{n}}$
$latex x_{1}x_{2}x_{3}+x_{1}x_{3}x_{4}+...+x_{2}x_{3}x_{4}+x_{2}x_{4}x_{5}+...+x_{n-2}x_{n-1}x_{n}=-\frac{a_{n-3}}{a_{n}}$
$latex ...\ dan\ seterusnya$
$latex x_{1}x_{2}x_{3}\ .\ ...\ .\ x_{n-1}x_{n}=(-1)^n\ .\ \frac{a_{0}}{a_{n}}$

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Contoh 1:
Persamaan kuadrat $latex x^{2}+5x-7=0$ memiliki akar-akar $latex x_{1}$ dan $latex x_{2}$
Tentukan nilai dari $latex {x_{1}}^3+{x_{2}}^3$.
Jawaban:
$latex n=2,\ a_{n}=a_{2}=1,\ a_{n-1}=a_{1}=5,\ a_{0}=-7$
$latex x_{1}+x_{2}=-\frac{a_{1}}{a_{2}}=-\frac{5}{1}=-5$
$latex x_{1}x_{2}=+\frac{a_{0}}{a_{2}}=+\frac{-7}{1}=-7$

sehingga diperoleh
$latex \begin{array}{rcl}{x_{1}}^3+{x_{2}}^3&=&(x_{1}+x_{2})^3-3{x_{1}}^2x_{2}-3x_{1}{x_{2}}^2\\&=&(x_{1}+x_{2})^3-3x_{1}x_{2}(x_{1}+x_{2})\\&=&(-5)^3-3(-7)(-5)\\&=&-125-105\\&=&-230\end{array}$


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Contoh 2:
Persamaan sukubanyak $latex x^{4}-5x^3-16x^2+41x-15=0$ memiliki akar-akar $latex a,\ b,\ c,$ dan $latex d$
Tentukan nilai dari: 
a. $latex a^2+b^2+c^2+d^2$
b. $latex \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$
Jawaban:
$latex n=4\\a_{n}=a_{4}=1,\ a_{n-1}=a_{3}=-5,\ a_{n-2}=a_{2}=-16,\ a_{n-3}=a_{1}=41,\a_{0}=-15$
$latex a+b+c+d=-\frac{a_{3}}{a_{4}}=-\frac{-5}{1}=5$
$latex ab+ac+ad+bc+bd+cd=+\frac{a_{2}}{a_{4}}=+\frac{-16}{1}=-16$
$latex abc+abd+acd+bcd=-\frac{a_{1}}{a_{4}}=-\frac{41}{1}=-41$
$latex abcd=+\frac{a_{0}}{a_{4}}=+\frac{-15}{1}=-15$

sehingga diperoleh 
jawaban a: 
$latex \begin{array}{rcl}a^2+b^2+c^2+d^2&=&(a+b+c+d)^2-2(ab+ac+ad+bc+bd+cd)\\&=&(5)^2-2(-16)\\&=&25+32\\&=&57\end{array}$
jawaban b: 
$latex \begin{array}{rcl}\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}&=&\frac{abc+abd+acd+bcd}{abcd}\\\\&=&\frac{-41}{-15}\\\\&=&\frac{41}{15}\end{array}$


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Contoh 3:
Jika $latex \alpha ,\ \beta ,$ dan $latex \gamma $ adalah akar-akar dari $latex x^{3}-x-1=0$, tentukan nilai dari $latex \frac{1+\alpha }{1-\alpha }+\frac{1+\beta }{1-\beta }+\frac{1+\gamma }{1-\gamma }$
Jawaban:
$latex n=3\\a_{n}=a_{3}=1,\ a_{n-1}=a_{2}=0,\ a_{n-2}=a_{1}=-1,\ \a_{0}=-1$
maka
$latex \alpha +\beta +\gamma =-\frac{a_{2}}{a_{3}}=-\frac{0}{1}=0$
$latex \alpha \beta +\alpha \gamma +\beta \gamma =+\frac{a_{1}}{a_{3}}=+\frac{-1}{1}=-1$
$latex \alpha \beta \gamma =-\frac{a_{0}}{a_{3}}=-\frac{-1}{1}=1$

sehingga diperoleh
$latex \begin{array}{rcl}\frac{1+\alpha }{1-\alpha }+\frac{1+\beta }{1-\beta }+\frac{1+\gamma }{1-\gamma }&=&\frac{(1+\alpha )(1-\beta )(1-\gamma )+(1+\beta )(1-\alpha )(1-\gamma )+(1+\gamma )(1-\alpha )(1-\beta )}{(1-\alpha )(1-\beta )(1-\gamma )}\\\\&=&\frac{3-(\alpha +\beta +\gamma )-(\alpha \beta +\alpha \gamma +\beta \gamma )+3\alpha \beta \gamma }{1-(\alpha +\beta +\gamma )+(\alpha \beta +\alpha \gamma +\beta \gamma )-\alpha \beta \gamma }\\\\&=&\frac{3-(0)-(-1)+3.1}{1-(0)+(-1)-1}\\\\&=&\frac{7}{-1}\\\\&=&-7\end{array}$

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