SOLUTION: What is the sum of the irrational roots of x^4+x^3-5x^2-3x+6=0?

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Question 1003820: What is the sum of the irrational roots of x^4+x^3-5x^2-3x+6=0?
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E4%2Bx%5E3-5x%5E2-3x%2B6=0

The potential rational roots are ± the factors of the
last term (in absolute value).

±1, ±2, ±3, ±6

We try 1

1 |1  1 -5 -3  6
  |   1  2 -3 -6
   1  2 -3 -6  0

So we have factored the equation as:

%28x-1%29%28x%5E3%2B2x%5E2-3x-6%29=0

We can factor the expression in the second parentheses
by grouping, ie., factor the first two terms and factor
the last two terms:

%28x-1%29%28x%5E2%28x%2B2%29-3%28x%2B2%29%29=0

Factor (x+2) out of the big parentheses:

%28x-1%29%28x%2B2%29%28x%5E2-3%29=0

Use zero-factor property:

x-1=0; x%2B2=0; x%5E2-3=0
x=1;    x=-2;  x%5E2=3
                   x=%22%22+%2B-+sqrt%283%29

The irrational roots are sqrt%283%29 and -sqrt%283%29.

Their sum is 0.

Edwin