A really sad polynomial
But seriously, folks, this is the resulting polynomial when a root is found through synthetic substitution. For example, when you have the equation x3 + x2+ 2x + 2 = 0...
-1 | 1 0 2 0
So we know one of the roots is -1. Which is the same as saying we've factor
ed (x+1) out of x3
+ 2x + 2. The second factor is the depressed polynomial. The order
of the polynomial is one less than the original, and the coefficient
s are equal to the results of the synthetic substitution, ignoring the last 0 which identifies it as a root. So in this case, it's x2
+ 0 x + 2, or x2
+ 2. So the entire equation can be rewritten as (x + 1)(x2
+ 2) = 0. We also know that the other two roots are complex because the second factor can't be factored any further.
Wouldn't you be sad if one of your factors was sucked out?