For the equation 2x 3 - 11x 2 + 12x + 9 = 0, the root 3 has a multiplicity of _____.

Your equation

2x^3 -11x^2 +12x +9 = 0
Can be factored to read
(x-3)^2 *(2x+1) = 0
which means 3 is a double root.

Please use a ^ before exponents when you write equations here.

To determine the multiplicity of a root, we need to observe the behavior of the equation near that specific root.

To find the multiplicity of the root 3 for the given equation, we can follow these steps:

Step 1: Identify the equation
The given equation is: 2x^3 - 11x^2 + 12x + 9 = 0

Step 2: Rewrite the equation in factored form
To determine the multiplicity of a root, it is helpful to rewrite the equation in factored form. For this, we need to find the factors of the equation. In this case, to factor the equation 2x^3 - 11x^2 + 12x + 9 = 0, we can use an algebraic method such as long division or synthetic division. However, the given equation seems to be challenging to factor using simple methods. Therefore, we can use numerical methods like approximation or graphical methods to locate the roots.

Step 3: Approximate the solution for the equation
We can use numerical approximation, such as the Newton-Raphson method or a graphing calculator, to find an approximate solution for the equation 2x^3 - 11x^2 + 12x + 9 = 0.

By applying the approximate method, we find that one of the roots is approximately x = 3.

Step 4: Observe the behavior of the equation near the root
Now, we need to observe the behavior of the equation near the root x = 3 to determine its multiplicity. We do this by checking the signs of the coefficients of the terms in the factored form of the equation or by investigating the behavior of the graph near the root.

By checking the signs of the coefficients of the terms, we observe the following:

- The term 2x^3 has a positive coefficient, indicating that the graph of the equation approaches the x-axis from the bottom left and continues upwards.
- The term -11x^2 has a negative coefficient, indicating the graph of the equation approaches the x-axis from the top right and continues downwards.
- The term 12x has a positive coefficient, indicating the graph of the equation approaches the x-axis from the bottom left and continues upwards.
- The constant term 9 is positive.

Step 5: Determine the multiplicity of the root.
Based on the observations made, we can conclude that the root 3 has a multiplicity of 1.