Problem

Solve the polynomial equation by factoring, and check the solutions graphically. \[ x^{3}-9 x=0 \] Rewrite the equation in a completely factored form. \[ x(x-3)(x+3)=0 \] (Type your answer in factored form.) The solution set is $\{\square\}$. (Simplify your answer. Use a comma to separate answers as needed.)

Solution

Step 1 :Given the polynomial equation \(x^{3}-9 x=0\).

Step 2 :Rewrite the equation in a completely factored form: \(x(x-3)(x+3)=0\).

Step 3 :The roots of the equation are the solutions to the equation. Set each factor equal to zero and solve for x: \(x=0\), \(x-3=0\), \(x+3=0\).

Step 4 :Solving these gives the solutions \(x=-3\), \(x=0\), \(x=3\).

Step 5 :Check the solutions graphically by plotting the equation and observing where it intersects the x-axis. The points of intersection are the solutions to the equation.

Step 6 :The graph of the equation \(y = x^3 - 9x\) intersects the x-axis at -3, 0, and 3, which are the solutions we found earlier. This confirms that the solutions are correct.

Step 7 :Final Answer: The solution set is \(\boxed{-3, 0, 3}\).

From Solvely APP
Source: https://solvelyapp.com/problems/YJzQ4Ao6f7/

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