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To solve the inequality involving the cubic polynomial \(2x^3 - 9x^2 + 4x \geq 0\), it's beneficial to start by factoring the polynomial. Factoring transforms the complex polynomial into a series of simpler expressions, which makes it easier to find solutions or zeros where the inequality equals zero.

When factoring polynomials, look for common factors first. In our cubic polynomial, you can factor out \(x\), which gives us \(x(2x^2 - 9x + 4)\). From here, finding the roots of \(2x^2 - 9x + 4\) can be complex, and it's helpful to use tools like the Rational Root Theorem or synthetic division. However, for polynomials that do not factor easily like this one, using a graphing calculator can help identify approximate real zeros instead.

A graphing calculator is a handy tool for understanding the behavior of complex polynomials like \(2x^3 - 9x^2 + 4x\). It assists in visualizing the roots and understanding where the polynomial is above or below the x-axis.

By inputting the polynomial into a graphing calculator, you can observe the graph's intercepts with the x-axis, giving a clear visual cue of the real zeros. For our polynomial, the graph indicates zeros at approximately \(x = 0\), \(x = 0.5\), and \(x = 4\). Understanding where a polynomial crosses or touches the x-axis helps to determine where it is positive or negative, crucial for solving inequalities.

Besides aiding in locating zeros, a graphing calculator simplifies complex calculations and saves time, especially when handling polynomials with higher degrees or coefficients.

Interval testing is an essential step in determining where a polynomial inequality, such as \(2x^3 - 9x^2 + 4x \geq 0\), holds true. Once you've identified the roots, they can be plotted on a number line, dividing the line into distinct intervals. For this polynomial, the intervals are \((-\infty, 0)\), \((0, 0.5)\), \((0.5, 4)\), and \((4, \infty)\).

Next, choose test points within each interval. Substituting these points into the polynomial shows whether the polynomial's value is positive or negative in that interval. For instance:

• For \(x = -1\), the calculation yields a negative value.
• For \(x = 0.25\), the result is positive, indicating the polynomial is positive in this interval.
• This process confirms the intervals over which the polynomial remains greater or equal to zero.

This method provides a systematic approach to ensuring the inequality holds over the correct intervals.

The real zeros of a polynomial represent the x-values where the polynomial equals zero. These zeros are crucial for solving polynomial inequalities because they help define the critical points on the number line.

For the polynomial \(2x^3 - 9x^2 + 4x\), the real zeros are \(x = 0\), \(x = 0.5\), and \(x = 4\). These are found using a combination of calculator visualization or factoring. Understanding these zeros helps you identify the solution set for the inequality. In this problem, the polynomial equals zero at these points, which are included in the solution set because the inequality is non-strict (\(\geq\)).

By including these zeros, the solution is extended to intervals that touch these points, turning the final solution into \([0, \infty)\). Real zeros help structure the analysis and ensure each step of solving the inequality is considered with precision.