root from a factor

Once you have factored an equation into a product equal to zero, each factor hands you a solution almost for free. A root from a factor is the value you get by setting one factor equal to zero and solving that small equation. This is the payoff of factoring: a hard polynomial equation collapses into a few one-step problems.

The reasoning rests on the zero-product property. If (x - 2)(x + 5) = 0, then either x - 2 = 0 or x + 5 = 0, so the roots are x = 2 and x = -5. Each linear factor (x - r) corresponds directly to the root x = r — the number that makes that factor vanish. This is why a factored polynomial wears its roots on its sleeve.

Two notes for accuracy. A repeated factor like (x - 3)^2 gives the same root twice — a 'double root' — which is a single number counted with multiplicity two. And a factor must equal zero, not some other value: only when the product is set to 0 does each factor independently produce a root.