Verify if polynomial (x^2+x-1)^(4m+1) -x may be divided by x^2-1?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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First, we notice that the difference of 2 squares x^2 - 1 could be written as a product of 2 factors.

x^2 - 1 = (x-1)(x+1)

If the given polynomial is divisible by x^2 - 1, then x = 1 and x = -1 are the roots of the polynomial.

We'll apply reminder theorem:

P(-1) = 0 (the reminder is 0 if x = -1 is a root)

P(1) = 0

We'll verify if P(-1) is cancelling if we'll substitute x by -1 in the expresison of polynomial:

P(-1) = [(-1)^2 - 1 - 1]^(4m+1) - (-1)

P(-1) = (1 - 2)^(4m+1) + 1

P(-1) = (- 1)^(4m+1) + 1

(-1) raised to an odd power yields -1.

P(-1) = -1 + 1 = 0

Therefore, x = -1 is a root of P(x).

We'll verify if P(1) is cancelling if we'll substitute x by 1 in the expresison of polynomial:

P(1) = [(1)^2 + 1 - 1]^(4m+1) - (1)

P(1) = (1)^(4m+1) - (1)

P(1) = 1 - 1

P(1) = 0

Since P(1) = 0, then x = 1 is also the root of P(x).

Since x = 1 and x = -1 are the roots of P(x), the polynomial (x^2+x-1)^(4m+1) - x is divisible by x^2 - 1.

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