depressed polynomial

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_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 x³ + x²+ 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 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?