Prove that x-1 is a factor of xn-1 for every natural number n-

Factor theorem says that for the polynomial p(x) and for the number a, if we have p(a) = 0, then (x minus; a) is a factor of p(x). Given: Polynomial xn m ...

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Mathematics

Factor Theorem

Prove that x-...

Question

Prove that x − 1 is a factor of xⁿ − 1 for every natural number n.

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Solution

Factor theorem says that for the polynomial p(x) and for the number a, if we have p(a) = 0, then (x − a) is a factor of p(x).

Given: Polynomial xⁿ− 1

Divisor = (x − 1)

Putting x = 1 in the given polynomial:

(1)ⁿ− 1

= 1 − 1 { (1)ⁿ = 1, for all natural numbers n}

= 0

∴( x − 1) is a factor of the polynomial xⁿ− 1 for every natural number n.

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Prove that x − 1 is a factor of xn − 1 for every natural number n.

Prove that x − 1 is a factor of xⁿ − 1 for every natural number n.