how would i solve this using the fundamental theorem of algebra?

x^3-10x^2+21x -108 = 0

Question

how would i solve this using the fundamental theorem of algebra? 

x^3-10x^2+21x -108 = 0

phi · Answer

The theorem tells you that there are 3 factors See http://www.mathsisfun.com/algebra/fundamental-theorem-algebra.html you would factor this polynomial. cubics are difficult.... but in this case, try (x-9) divide (x-9) into the polynomial to factor it.

anonymous · Answer

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phi · Answer

yes,that will work

anonymous · Answer

what would be the Rational Root Theorem?

phi · Answer

See http://www.mathwords.com/r/rational_root_theorem.htm but as you see, 108 will give lots of possibilities...

anonymous · Answer

how would i know which one is q and p?

phi · Answer

p is one of the factors of the last number -108  and q is the number in front of x^3  (1, which is not usually written because 1*x^3 is x^3)

your possible roots will be ± p/q.  q is 1, so just ±p
now you list all the numbers that divide into 108:
1,2,3, 4, 6, 9, 12, ....  (there are lots of them)

phi · Answer

next time, pick a smaller box!

anonymous · Answer

well, i thought is was pretty small, i didnt even use a box cuz i wanted smaller, so i used a book

anonymous · Answer

i have to list ALL #´s that divide into 108?

phi · Answer

that is what the theorem says...  that the root you want is among those numbers.
Of course 9 is listed, so it works, but all the other numbers are not roots.  
If you tested each one, smallest to biggest (by dividing x-a  where a is one of the numbers and x+a  (could be either + or -) )  you could say, after finding 9,  I don't need the rest of the factors of 108....

phi · Answer

Or you could say you are like the rainman and can do these things in your head.. it's 9!

anonymous · Answer

i got it, thanks!

anonymous · Answer

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phi · Answer

one way to find the roots is start with your list
±1,±2,±3, ±4, ±6, ±9, ±12, ....  
replace x with each possible root:  for example x= -1
x^3-10x^2+21x -108 = 0
-1  -10 -21 -108  ≠ 0   and see if you get zero.  no with this one
with x=1:
1-10+21-180  No
keep going until you get to x=+9
729-810+189-108=0  pay dirt

once you know 9 is a root, you can divide (as you did up above) to get 
x^2-x+12=0
use the quadratic formula to show that you do not get any real answers.
so x=9 is the  only real solution.