How do you find all the zeros of # f(x)= x^4+x^3+2x^2+4x-8# with its multiplicities?
First, we use the rational root theorem to try and pick out a possible root. I always try 1 first because it is easy to do: just count the sum of the coefficients. Luckily enough, it worked!
So now we synthetically divide
x1 1 1 2 4 -8
1 2 4 8
result1 2 4 8 0
so now we have
Applying guess and check work for the rational root theorem again, we find that
So we go back to synthetic division
x-2 1 2 4 8
-2 0 -8
Result 1 0 4 0
Now we have
We use the quadratic formula to find that the other two roots are
So finally, our zeroes are