Universal set,
`U={x|x^(5)-6x^(4)+11x^(3)-6x^(2)=0}`
`A={x|x^(2)-5x+6=0}`
`B={x|x^(2)-3x+2=0}`
What is `(AnnB)`' equal to ?

Options

A. {1, 3}
B. {1, 2, 3}
C. {0, 1, 3}
D. {0, 1, 2, 3}

Explanation

To solve the problem step by step, we need to determine the elements of the universal set \( U \), the sets \( A \) and \( B \), and then find the complement of the intersection of sets \( A \) and \( B \).

### Step 1: Find the Universal Set \( U \)

The universal set is defined by the equation:
\[ x^5 - 6x^4 + 11x^3 - 6x^2 = 0 \]
To find the roots, we can factor the polynomial.

Factoring out \( x^2 \):
\[ x^2(x^3 - 6x^2 + 11x - 6) = 0 \]
This gives us one root \( x = 0 \). Now we need to solve the cubic equation:
\[ x^3 - 6x^2 + 11x - 6 = 0 \]
Using the Rational Root Theorem, we can test possible rational roots (1, 2, 3, etc.) and find that \( x = 1, 2, 3 \) are roots.

Thus, we can factor the cubic polynomial as:
\[ (x - 1)(x - 2)(x - 3) = 0 \]
So the complete factorization of the original polynomial is:
\[ x^2(x - 1)(x - 2)(x - 3) = 0 \]
The roots are \( x = 0, 1, 2, 3 \).

**Universal Set \( U \)**:
\[ U = \{0, 1, 2, 3\} \]

### Step 2: Find Set \( A \)

Set \( A \) is defined by the equation:
\[ x^2 - 5x + 6 = 0 \]
Factoring gives:
\[ (x - 2)(x - 3) = 0 \]
Thus, the roots are:
\[ x = 2, 3 \]
**Set \( A \)**:
\[ A = \{2, 3\} \]

### Step 3: Find Set \( B \)

Set \( B \) is defined by the equation:
\[ x^2 - 3x + 2 = 0 \]
Factoring gives:
\[ (x - 1)(x - 2) = 0 \]
Thus, the roots are:
\[ x = 1, 2 \]
**Set \( B \)**:
\[ B = \{1, 2\} \]

### Step 4: Find \( A \cap B \)

The intersection \( A \cap B \) consists of elements that are in both sets \( A \) and \( B \):
\[ A \cap B = \{2\} \]

### Step 5: Find \( (A \cap B)' \)

The complement of the intersection \( (A \cap B)' \) is found by taking the universal set \( U \) and removing the elements in \( A \cap B \):
\[ (A \cap B)' = U - (A \cap B) = \{0, 1, 2, 3\} - \{2\} = \{0, 1, 3\} \]

### Final Answer

Thus, the result for \( (A \cap B)' \) is:
\[ (A \cap B)' = \{0, 1, 3\} \]

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