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

since we know 3 is a roots, x-3 must be a factor, so

2x^3 - 11x^2 + 12x + 9
= (x-3)(2x^2 - 5x - 3)
= (x-3)(x-3)(2x+1)

so .....

To determine the multiplicity of a root in a polynomial equation, we need to look at the degree of the root's factor in the equation.

In this case, the given equation is 2x^3 - 11x^2 + 12x + 9 = 0.

To determine the multiplicity of the root 3, we need to factorize the equation and check the degree of the root's factor.

We can start by testing if the root 3 is a factor of the equation by substituting x = 3 into the equation:

2(3)^3 - 11(3)^2 + 12(3) + 9 = 0
54 - 99 + 36 + 9 = 0
0 = 0

Since the result is 0, it confirms that x - 3 is a factor of the equation.

Now, divide the given equation by (x - 3) to find the other factor:

(2x^3 - 11x^2 + 12x + 9) ÷ (x - 3)

Performing polynomial long division or synthetic division, we find that the other factor is 2x^2 + 5x + 3.

Therefore, the root 3 has a multiplicity of 1 in the given equation.

To determine the multiplicity of a root, we need to factor the equation and observe how many times the root appears in the factors.

First, let's write the equation in factored form. We use the root 3 to factor the equation:

(x - 3)(2x^2 + 7x - 3) = 0

Now, let's focus on the factors. We can see that the root 3 appears once in the factor (x - 3) and does not appear in the factor (2x^2 + 7x - 3).

Therefore, the root 3 has a multiplicity of 1.