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To fully understand where the function changes its sign, we need to find its critical points. Critical points occur where the derivative of the function is zero or undefined. For the cubic polynomial given, calculate the derivative \(f'(x)\). This derivative will help identify where the slope of the function's graph is zero, implying a local maximum or minimum. In our case:- \(f(x) = -x^3 + 3x^2 + 10x\)- \(f'(x) = -3x^2 + 6x + 10\)Setting the derivative equal to zero, we solve \(-3x^2 + 6x + 10 = 0\) to find the critical points.
By determining these critical points, we can better understand the graph's behavior, particularly how the function transitions from increasing to decreasing, or vice versa.

Quadratic equations appear when working with cubic polynomials due to factoring down higher-degree polynomials. Here, solving \(-x^2 + 3x + 10 = 0\) is essential for determining the roots of the cubic equation.
We use the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Where, for our quadratic equation, \(a = -1\), \(b = 3\), and \(c = 10\).- Calculating the discriminant, \(b^2 - 4ac = 9 + 40 = 49\).- Then plug into the quadratic formula,- This results in two solutions: \(x = -2\) and \(x = 5\).
Understanding how to solve quadratic equations facilitates finding roots of polynomials, especially when dealing with factored terms.

After finding the roots of the polynomial through critical points and solving quadratic equations, the next step is determining intervals of positivity and negativity. These intervals help us understand where the function is above or below the x-axis. By setting the function \(f(x)\) into parts with the roots as boundaries, \([-\infty, -2], [-2, 0], [0, 5], [5, \infty]\), we can test interval signs by evaluating the function at any point in these ranges:

• Choosing \(x = -3\) in \([-\infty, -2]\) gives a positive sign.
• Choosing \(x = -1\) in \([-2, 0]\) also yields positive.
• Choosing \(x = 1\) in \([0, 5]\) remains positive.
• Choosing \(x = 6\) in \([5, \infty]\) shows a negative sign.

These test points confirm when \(f(x)\) is above or below the x-axis, providing clear intervals for graphing and function analysis.

Graph sketching for a cubic polynomial involves using roots, critical points, and intervals of positivity/negativity. Begin by plotting the roots at \(-2, 0,\) and \(5\) on the x-axis. These are intersections where the graph meets the x-axis.
Secondly, use the identified intervals:

• For \([-\infty, -2], [-2, 0],\) and \([0, 5]\), \(f(x) > 0\), showing the graph lies above the x-axis.
• For \([5, \infty]\), the graph dips below the x-axis since \(f(x) < 0\).

The graph sketch becomes clearer by connecting these plotted points and ensuring the transition through each critical point and direction of slope is correctly represented. Sketching reflects all the behavior identified in analysis, illustrating the turning points and end behavior effectively.

The roots of a polynomial are the x values that make \(f(x) = 0\). Finding these for a cubic polynomial like \(f(x) = -x^3 + 3x^2 + 10x\) is crucial as they indicate where the graph crosses the x-axis.
To determine the roots, factor or apply methods like solving related quadratic equations:- Factoring \(x(-x^2 + 3x + 10) = 0\), \(x = 0\) is a root.- Solving the quadratic part \(-x^2 + 3x + 10 = 0\) gives roots \(x = -2\) and \(x = 5\) through the quadratic formula.
Thus, the roots are \(x = -2, 0,\) and \(5\). Identifying these helps in predicting the behavior of the graph, ensuring an accurate depiction of where the cubic polynomial will change direction or cross the x-axis.