Graph the distinct real zeros of the polynomial function $p(x)=x^3-2 x^2-4 x^2+8 x$.
We are given the polynomial function \(p(x) = x^3 - 2x^2 - 4x^2 + 8x\). To find its zeros, we first simplify the expression by combining like terms. Adding \(-2x^2\) and \(-4x^2\) gives:
\[p(x) = x^3 - 6x^2 + 8x\]
To find the zeros, we solve the equation \(p(x) = 0\):
\[x^3 - 6x^2 + 8x = 0\]
We factor out the common term \(x\) from the equation \(x^3 - 6x^2 + 8x = 0\):
\[x(x^2 - 6x + 8) = 0\]
Next, we factor the quadratic trinomial \(x^2 - 6x + 8\). We look for two numbers that multiply to \(8\) and add to \(-6\). These numbers are \(-2\) and \(-4\). Thus:
\[x^2 - 6x + 8 = (x - 2)(x - 4)\]
The fully factored form of the polynomial is:
\[x(x - 2)(x - 4) = 0\]
The zeros occur where the factored polynomial equals zero. Setting each factor to zero:
The distinct real zeros of the polynomial are \(x = 0\), \(x = 2\), and \(x = 4\). These are the points on the x-axis where the graph of the function intersects the axis.
In physics, finding the zeros of a polynomial can represent the times when an object's position is at the origin or when its velocity is zero. For example, if the position of an object is modeled by \(p(t) = t^3 - 6t^2 + 8t\), then solving for the zeros tells us the specific times \(t=0, 2, 4\) seconds when the object passes through the starting position. This helps in understanding the motion trajectory of projectiles or vehicles.