The equation 12x^4 + 20x^3 - 11x^2 - 5x + 2 = 0 has one integer root. Find this root and hence, solve the equation completely.
[I found the integer root already that is x + 2, but I do not know how to solve it because I do not know the method in finding out factor that is in fraction. who can teach me with this please..?]

Question

The equation 12x^4 + 20x^3 - 11x^2 - 5x + 2 = 0 has one integer root. Find this root and hence, solve the equation completely. 
[I found the integer root already that is x + 2, but I do not know how to solve it because I do not know the method in finding out factor that is in fraction. who can teach me with this please..?]

anonymous · Answer

If  "completely" means "find all the roots" you will have to solve a cubic after you have factored out (x+2).

anonymous · Answer

yes. but I do not know the way in searching out the factors of the cubic, 12x^3 - 4x^2 - 3x + 1. one of its factors is 1/3, the other one is a complex number. do you know the step to find the factor that is a fraction?

anonymous · Answer

The integer root is -2 and the other roots are 1/2, -1/2, and 1/3.

anonymous · Answer

nope.. the answers are -2, 1/3 and + or - 1/2

anonymous · Answer

Use synthetic division, but before you do divide both sides of the equation by 12 so you have the leading coefficient equal to 1.

anonymous · Answer

hm.. what's synthetic division? After divide both side with 12, then should I find the factor of 1/12?

anonymous · Answer

i could help too

anonymous · Answer

now since you know that x=-2 is a root
for synthetic division write d coefficients of the equation as
   12 20 -11 -5  2

anonymous · Answer

The synthetic algorithm is fine for the first root because you have been given it but doesn't help for the rest because you don't know (officially) what the roots are, so you have to solve a cubic.

anonymous · Answer

$$  -2\lfloor 12     20    -11 -5   2     $$

anonymous · Answer

$$-2\lfloor 12      20       -11    -5       2$$

anonymous · Answer

-2 | 12 20 -11 -5 2
         0

anonymous · Answer

now write the diff or sum of the first term i.e
-2|  12 20 -11 -5 2
    |   0  
    |   12

anonymous · Answer

multiply d divisor with the remainder and write the no below the second term

anonymous · Answer

-2| 12 20 -11 -5 2
    |   0  24
    |   12

anonymous · Answer

sorry its -24

anonymous · Answer

as you keep doing this you finally get
-2 | 12 20 -11   -5    2
     |  0 -24    8    6     -2
     |  12  -4   -3   1     0

anonymous · Answer

As -2 is the root you get d last term as zero, now u hav 4 terms in d bottom most line.. i.e a 3rd order equation

anonymous · Answer

$$12x ^{3}-4x^{2}-3^{x}+1=0$$

anonymous · Answer

solve this.. take 4x^2 common in d first 2 terms

anonymous · Answer

$$4x ^{2}(3x-1)-1(3x-1)=0$$

anonymous · Answer

now take 3x-1 as common

anonymous · Answer

$$(3x-1)(4x ^{2}-1)=0$$

anonymous · Answer

from d first equation you get x=1/3

anonymous · Answer

from the second you get x=+-1/2