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 x^{3} + x^{2}+ 2x + 2 = 0...

___|_1_1_2_2___
-1 | 1 0 2 0

So we know one of the roots is -1. Which is the same as saying we've

factored (x+1) out of x

^{3} + x

^{2}+ 2x + 2. The second factor is the depressed polynomial. The

order of the polynomial is one less than the original, and the

coefficients 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 x

^{2} + 0 x + 2, or x

^{2} + 2. So the entire equation can be rewritten as (x + 1)(x

^{2} + 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?