Question

Help solve 4x^3-12x^2-x+3=0?

1) list all possible rational roots?
2) solve the equation.

Solve the equation x^3-8x^2+21x-20=0 given that one of the roots is 2+i

Answer 1

Well, do it. Factors of 3 divided by factors of 4. That Is ALL the possible Rational Roots.

With Real coefficients, Complex Roots must come in pairs. Consider 2-i, also.

Answer 2

i dont understand

Answer 3

Ok, you can get 4x^3-12x^2-x+3=0 into this form

4x^2(x-3)-(x-3)=0

4x^2(x-3)=x-3

4x^2=1

x^2=1/4

x=+- 1/2

real roots are +1/2 and -1/2

Answer 4

@cblrtopas is the 3rd root 3?

Answer 5

u know how to do synthetic division?

Answer 6

Yes actually, I think it is

Answer 7

it satisfies the equation so yes....

Answer 8

@dan815 yes
@cblrtopas okay, but i dont know how 3 was found. i didnt do that, someone told me.

Answer 9

@cblrtopas and also, when you worked out the steps, you put 4x^2 and its to the 3rd power.

Answer 10

well you look at the coefficient of 1st and last numbers
so 4x^3-12x^2-x+3=0?
so 4 and 3 you try multiples so 1 , 2 , 3 , 4 you find 3 works

Answer 11

I eliminated x=3 from the solution set by dividing by x-3 in my derivation earlier. When dividing by x-3 I was implicitly assuming that the quantity x-3 was not zero. In this case it's possible that x was indeed 3.

Basically, you can get the third root by using a cubic equation or some fancy factorization. I don't remember this crap from the top of my head I need to look at my algebra book.

Answer 12

then (x-3) syenthtic division gives you (x-3)(2x-1)(2x+1)

Answer 13

tessa11, you need to go through each step carefully. The 4x^2 is multiplied by (x-3). which will yield the cubic when multiplied out. Don't copy my work make the derivation yourself..

Answer 14

@cblrtopas thank you! i think i get it, but how would i do the 2nd equation? because it gives me a root, and it has an imaginary number in it.

Answer 15

what second equation?

Answer 16

@cblrtopas Solve the equation x^3-8x^2+21x-20=0 given that one of the roots is 2+i

Answer 17

You didn't copy the problem correctly I think (2-i) is not a root if there's no x somewhere in it.

Answer 18

@cblrtopas no thats what my teacher gave me exactly :(

Answer 19

Ugh......It's late.....do you even know what imaginary numbers are used for...

Seriously screw imaginary algebra .....it will never become useful

I don't know girl...

Answer 20

@cblrtopas haha alright, well thank you for your help!!! :)

Answer 21

\[(x-3) (2 x-1) (2 x+1)=0 \]