Polynomial Equations Written in Factored Form

The zero product property says a product of numbers equals zero only when one of them is zero. Factored form turns a polynomial into exactly that kind of product. Write a polynomial as (x - r1)(x - r2)...(x - rn) and the equation (x - r1)(x - r2)...(x - rn) = 0 is true whenever any single factor is zero. That gives you every real solution instantly, without expanding, dividing, or guessing. This works for two factors or twenty: scale doesn't change the logic, it just means checking each factor once.

A repeated factor like (x - 4)² still represents one condition: x - 4 = 0, so x = 4. Squaring it doesn't create a second, different solution, it just means that factor vanishes using two identical copies of the same root. So x(x - 4)²(x + 3) = 0 has solutions x = 0, x = 4, and x = -3, with x = 4 coming from a repeated factor. The distinct solutions are still just three numbers.

This connects directly to graphing: a factor appearing once makes the graph cross the x-axis there, while a squared factor makes the graph touch the x-axis and bounce back, since the sign of that factor doesn't change. Going the other direction, if you're told the zeros, you can rebuild the factored form by writing one factor per zero, repeating a factor for any repeated zero. That link between zeros, factors, and the shape of the graph is the core skill behind this whole family of problems.