Prove that x-1 is a factor of xn-1 for every even 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 even 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)

To check whether (x + 1) is a factor of xⁿ− 1, we have to convert (x + 1) in a form that is suitable for the application of the Factor theorem.

(x + 1) = {x − (−1)}

Putting x = −1 in the given polynomial:

(−1)ⁿ− 1

= 1 − 1 { (−1)ⁿ = 1, for every even number n}

= 0

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

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

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