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Explain to me wether (x-2) is a factor of x^3 -x +8 and Why
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Use the Factor Theorem: (x − a) is a factor of a polynomial P(x) iff P(a) = 0.
Here P(x) = x^3 − x + 8. Evaluate at x = 2:
P(2) = 2^3 − 2 + 8 = 8 − 2 + 8 = 14 ≠ 0.
Therefore (x − 2) is not a factor. (Synthetic division by x − 2 gives quotient x^2 + 2x + 3 and remainder 14, since x^3 − x + 8 = (x − 2)(x^2 + 2x + 3) + 14.)
Here P(x) = x^3 − x + 8. Evaluate at x = 2:
P(2) = 2^3 − 2 + 8 = 8 − 2 + 8 = 14 ≠ 0.
Therefore (x − 2) is not a factor. (Synthetic division by x − 2 gives quotient x^2 + 2x + 3 and remainder 14, since x^3 − x + 8 = (x − 2)(x^2 + 2x + 3) + 14.)
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