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For all positive integers n and for all integers x > 1, xⁿ - 1 is divisible by x - 1?

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Final answer:

Yes, for all positive integers n and integers x > 1, the expression xⁿ - 1 is divisible by x - 1, as demonstrated by the remainder theorem, factor theorem, and the binomial theorem.

Step-by-step explanation:

The student's question addresses a concept in algebra, specifically the divisibility of expressions of the form xⁿ - 1 by x - 1. To determine if xⁿ - 1 is divisible by x - 1 for positive integers n and integers x > 1, we can apply the remainder theorem or use polynomial division. According to the remainder theorem, if a polynomial f(x) is divided by x - c, the remainder is f(c). Hence, if we substitute x = 1 into xⁿ, the result is 1, and 1 - 1 = 0; therefore, xⁿ - 1 is indeed divisible by x - 1. This is also a special case of the factor theorem, which states that x - c is a factor of the polynomial if f(c) = 0. Moreover, using the binomial theorem, we can expand xⁿ - 1 and factor out x - 1 to see the divisibility directly.

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