When expanding a factorised polynomial expression, the x^1 term will be a sum of the products of every linear constant but without a single constant. ie: When expanding (x+a)(x+b)(x+c)(x+d), the x^1 term has a coefficient that is abc+abd+acd+bcd. This kind of calculation feels like one that has a tricksy shortcut using algebra, number theory, or linear algebra, but I can't think of any way to make it faster. The only thing I thought might have some merit is computing z=abcd then calculating z/d + z/c + z/b + z/a.
Is there some kind of trick or workaround that allows one to find this coefficient that is better than just working it out straightforwardly?