Answers

The Brainliest Answer!
2014-10-01T04:55:53+05:30

This Is a Certified Answer

×
Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
3x^3-x^2-8x-2=0 \\ \\

Trying with rational factors of 2/3 - does not give us a root.  This equation has irrational roots.
We reduce the equation by the following method.

Let a = 3,  b = -1,   c = -8   ,   d = -2

Substitute y = x + b/3a       ie., y = x -1/9

You will get 
y^3 - \frac{73}{27}y-\frac{704}{729} = 0,\ \ \ or, \ \ \ y^3-\frac{73}{3^3}y-\frac{11*2^6}{3^6} = 0 \\ \\ let\ \ p = -\frac{73}{3*27}=-0.90123,\ \ \ \ \ \ q=704/729 = 0.9657 \\ \\Solution\ for\ y^3-3p\ y-q= 0,\ \ is\ given\ by\\ \\ y_k, \ for\ k=0,1,2,\ \ is = 2\sqrt{p}\ *\ Cos[ \frac{1}{3}*\ Cos^{-1}(\frac{q}{2p^{\frac{3}{2}}} ) - k\ *\ \frac{2 \pi }{3} ] \\ \\y_0 = 1.800005105\\y_1=-0.37699\\y_2=-1.42305\\ \\ x = y +\frac{1}{9}\\ \\ x=1.911164,\ \ \ -0.26588,\ \ \ -1.31194\\

We follow the above when p is positive in reduced cubic equation.  If p is negative in the reduced cubic equation then we do the following.  Otherwise, we have work with complex roots. It becomes difficult to evaluate roots.

y^3+py-q=0\\ \\Substitute\ y=s-\frac{p}{s},\ \ to\ get\ \ s^6-q\ s^3-p^3=0\\ \\ s^3=\frac{q}{2}+-\frac{1}{2}\sqrt{q^2-4*p^3}\\ \\t^3=-\frac{q}{2}+-\frac{1}{2}\sqrt{q^2-4*p^3}\\ \\Find\ s\ and\ t\ from\ s^3\ and\ t^3.\\ \\ y = s - t\\ find \\ x = y - b/3a\\

2 5 2
hope it is not too difficult to understand. but cubic equation given is not simple one.
thanx n u r welcom