# Find all solutions for....

• Dec 8th 2008, 06:23 AM
calvin_coolidge
Find all solutions for....
3tan^3x-3tan^2x-tanx+1=0

I started by grouping/factoring out tanx...giving me tanx(3tan^2x-3tanx-1)+1=0...can't factor it...stuck @ the moment...

Any thoughts or a push in the right direction would be appreciated!

Thx,

~CC.
• Dec 8th 2008, 06:37 AM
samer_guirguis_2000
This is a polynomial of the third degree which can be solved as follows:
let tan(x) =y
3y^3-3y^2-y+1=0
dividing the equation by 3 (the coefficient of the highest power)
y^3-y^2-y/3+1/3
the last term (+1/3) is the product of the three roots of the equation. This means that the roots should contain the value 1.
dividing the equation by (y-1)
then the above equation will be
(y-1)(3y^2-1)=0
the second bracket can be solved by the known equation of solving quadratic equations
the root are 1,1/root(3),-1/root(3)
• Dec 8th 2008, 06:37 AM
Chop Suey
Let $u = \tan{x}$

$3u^3-3u^2-u+1=0$

$\implies 3u^2(u-1)-(u-1)=0$

$\implies (3u^2-1)(u-1)=0$

EDIT: Ah, Samer beat me to it. And for some reason, I can't delete my post. O_O
• Dec 8th 2008, 06:41 AM
Quote:

Originally Posted by calvin_coolidge
3tan^3x-3tan^2x-tanx+1=0

Maybe you can try this

Let $y = \tan x$ , thus you will have $3y^3-3y^2-y+1=0$
Then solve for y and you should be able to find the solutions .
• Dec 9th 2008, 06:56 AM
calvin_coolidge
Thank you all for your assistance!

~CC
• Dec 9th 2008, 03:07 PM
Crazyheaven
Not to hijack the thread but could someone explain this step for me.

"the last term (+1/3) is the product of the three roots of the equation. This means that the roots should contain the value 1.
dividing the equation by (y-1)"

I'm not really understanding why that is and when to use it.
• Dec 10th 2008, 04:07 AM
samer_guirguis_2000
Any equation of the third degree can be written in two forms:
1- y^3+ay^2+by+c=0
2- (y-y1)*(y-y2)*(y-y3)=0
where y1,y2,y3 are the roots of the equation
if you multiply these three brackets you will find that (-y1y2y3)=c provided that the coeffecient of y^3 is 1