1. ## Find all roots...

Uhm, hi again you guys. ^^" I have root problems, it seems.
And other problems. (I think I'm going to be posting up more problems because I have forgotten a lot of the stuff I've learned. This is like a review extra credit assignment.)

Find all the roots of the equation f(x) = 6x^5-13x^4+27x^3+22x^2-78x+36=0
It suggests using Rational Root Theorem, Descartes rule, and such. I only managed to find two roots...they are 1 and 2/3. I'm left with 6x^3-3x^2+18x+54=0 as the quotient...and when I try to continue trial and error with synthetic division, I can't find the other roots. Help please!
Thank you.

2. Originally Posted by Jukixel
Uhm, hi again you guys. ^^" I have root problems, it seems.
And other problems. (I think I'm going to be posting up more problems because I have forgotten a lot of the stuff I've learned. This is like a review extra credit assignment.)

Find all the roots of the equation f(x) = 6x^5-13x^4+27x^3+22x^2-78x+36=0
It suggests using Rational Root Theorem, Descartes rule, and such. I only managed to find two roots...they are 1 and 2/3. I'm left with 6x^3-3x^2+18x+54=0 as the quotient...and when I try to continue trial and error with synthetic division, I can't find the other roots. Help please!
Thank you.
By Descartes rule of signs there is exactly 1 negative root of this cubic, and a bit of trial and error (or the rational root theorem) suggests you try -3/2

.

3. Oh, wow. xD Thanks a lot! When I did Descartes rule, I accidentally changed an even power's sign and ended up with 2 variations. ^^" That confused me and then I thought that there weren't any negative roots because it could vary by an even integer.
Thank you!