Find all roots: $x(2x-5)(x-1)=0$
The equation provided is \(x(2x - 5)(x - 1) = 0\). This is a polynomial equation already given in its factored form, which is a product of three linear factors. Our goal is to find all values of \(x\) that satisfy this equation. The Zero Product Property states that if a product of factors is equal to zero, then at least one of the factors must be zero. Mathematically, for factors \(A, B,\) and \(C\), if \(A \cdot B \cdot C = 0\), then \(A=0\) or \(B=0\) or \(C=0\).
Applying the Zero Product Property, we set each of the three factors equal to zero independently:
Now, we solve each of the resulting equations for \(x\):
Adding 5 to both sides gives \(2x = 5\).
Dividing both sides by 2 gives \(x = \frac{5}{2}\) or \(x = 2.5\).
Adding 1 to both sides gives \(x = 1\).
Having solved all individual linear equations, we have found the roots of the original polynomial equation. The roots are \(x = 0\), \(x = 1\), and \(x = 2.5\). Thus, the set of all roots is \(\{0, 1, 2.5\}\). The final answer is \(\boxed{0, 1, 2.5}\).
The Zero Product Property is frequently used in physics and engineering when modeling motion. For example, if a projectile's height above the ground is given by a factored equation like \(h(t) = -t(t-4)(t-2)\), setting \(h(t) = 0\) helps identify the exact times \(t\) when the projectile is at ground level. Understanding these roots allows us to determine when an object launches, lands, or reaches specific milestones in its path.