How do I find the zeros of the polynomial function [math]f(x)=\dfrac{1}{2}x^{3}-3x[/math]?

To find the zeros of the polynomial function ( f(x) = \frac{1}{2}x^3 - 3x ), you can follow these steps, utilizing several methods as outlined below:

Step 1: Factor Out Common Terms

First, factor out the common term ( x ) from the polynomial: [ f(x) = \frac{1}{2}x^3 - 3x = x\left(\frac{1}{2}x^2 - 3\right) ]

Step 2: Set the Factored Expression Equal to Zero

Set the entire expression equal to zero to find the zeros: [ x\left(\frac{1}{2}x^2 - 3\right) = 0 ]

Step 3: Apply the Zero-Product Property

Using the zero-product property, set each factor equal to zero: [ x = 0 ] or [ \frac{1}{2}x^2 - 3 = 0 ]

Step 4: Solve the Quadratic Equation

For the quadratic equation ( \frac{1}{2}x^2 - 3 = 0 ), solve for ( x ): [ \frac{1}{2}x^2 = 3 ] [ x^2 = 6 ] [ x = \pm \sqrt{6} ]

Step 5: List All Zeros

Combine the results to list all the zeros of the polynomial: [ x = 0, \quad x = \sqrt{6}, \quad x = -\sqrt{6} ]

Alternative Methods

• Graphical Method: You can also find the zeros by plotting the function ( f(x) = \frac{1}{2}x^3 - 3x ) and identifying the points where the graph intersects the ( x )-axis. However, this method may not provide exact values as easily as algebraic methods135.

• Synthetic Division or Rational Root Theorem: These methods are more applicable when you have a polynomial with integer coefficients and are looking for rational zeros. In this case, since the polynomial simplifies to a form that can be easily solved algebraically, these methods are not necessary5.

By following these steps, you can accurately determine the zeros of the given polynomial function.