Sure! Let's solve this question step-by-step.
First, we need to understand what the question is asking. We are given that 11 is a rational root with multiplicity 2. A root of a polynomial is a value that satisfies the polynomial equation when substituted into it. The multiplicity of a root indicates how many times this root is repeated in the factorization of the polynomial.
Thus, a root of 11 with multiplicity 2 would be represented by the factor \((x - 11)^2\). Let's go through each of the given factors to determine if they represent this condition:
1. \((x - 11)^2\)
This directly matches the condition. It is a factor with a root of 11 and multiplicity 2.
2. \((x + 11)^2\)
This represents a root of -11 with multiplicity 2, not a root of 11.
3. \((x - 11)(x + 11)\)
This factorization represents two roots, 11 and -11, each with multiplicity 1. It does not satisfy the condition of multiplicity 2 for the root 11.
4. \((2x - 11)\)
This factor would represent a root of \( \frac{11}{2} \), not 11. Also, it does not have multiplicity 2.
5. \(2(x - 11)\)
This factor represents the same root of 11, but it is multiplied by 2. It is not an indication of the multiplicity of the root, and does not match the multiplicity 2 requirement.
Given the condition of having a rational root of 11 with a multiplicity of 2, the factor that satisfies this condition is:
[tex]\((x - 11)^2\)[/tex].
