1. **Stating the problem:** We are given a prime number $p$ and asked to find the quadratic equation whose factors are zero. 2. **Understanding the problem:** If the factors of a quadratic equation are zero, it means the roots of the quadratic equation are the values that make each factor zero. 3. **Formula used:** For a quadratic equation with roots $r_1$ and $r_2$, the equation can be written as: $$ (x - r_1)(x - r_2) = 0 $$ Expanding this, we get: $$ x^2 - (r_1 + r_2)x + r_1 r_2 = 0 $$ 4. **Applying the problem:** Since $p$ is a prime number, the factors that are zero could be $x = 0$ and $x = p$. 5. **Forming the equation:** Using roots $r_1 = 0$ and $r_2 = p$, the quadratic equation is: $$ (x - 0)(x - p) = 0 $$ Expanding: $$ x(x - p) = 0 $$ $$ x^2 - p x = 0 $$ 6. **Final answer:** The quadratic equation with factors zero at $0$ and $p$ is: $$ x^2 - p x = 0 $$