Mathematics

The Factor Theorem connects roots and factors: if a polynomial equals zero at some input, that input reveals a linear factor. When $p(2)=0$, you know the polynomial "touches zero" at $x=2$, which means $(x-2)$ must divide evenly into $p(x)$ with no remainder. Think of it as the polynomial having a built-in "zero-maker"—whenever you plug in 2, the whole expression collapses to 0, and that only happens if $(x-2)$ is lurking inside as a factor.

The Factor Theorem connects roots and factors: if a polynomial equals zero at some input, that input reveals a linear factor. When $p(2)=0$, you know the polynomial "touches zero" at $x=2$, which means $(x-2)$ must divide evenly into $p(x)$ with no remainder. Think of it as the polynomial having a built-in "zero-maker"—whenever you plug in 2, the whole expression collapses to 0, and that only happens if $(x-2)$ is lurking inside as a factor.