The set A = {x : x4 – x3 – x2 = 0 and x ∈ N} represents
Question
The set A = {x : x4 – x3 – x2 = 0 and x ∈ N} represents
Solution
The set A represents the natural numbers x that satisfy the equation x^4 - x^3 - x^2 = 0.
Step 1: We can factor out an x^2 from each term to simplify the equation: x^2(x^2 - x - 1) = 0.
Step 2: This equation is satisfied if either x^2 = 0 or x^2 - x - 1 = 0.
Step 3: The first equation, x^2 = 0, is satisfied by x = 0. However, 0 is not a natural number, so it is not included in the set A.
Step 4: The second equation, x^2 - x - 1 = 0, is a quadratic equation. We can solve it using the quadratic formula x = [1 ± sqrt(1 + 4)]/2.
Step 5: This gives us two solutions: x = [1 + sqrt(5)]/2 and x = [1 - sqrt(5)]/2. However, neither of these are natural numbers, so they are not included in the set A.
Therefore, the set A = {x : x^4 – x^3 – x^2 = 0 and x ∈ N} represents the empty set, as there are no natural numbers x that satisfy the given equation.
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