Is (x − 2) a factor of f(x) = x3 − 2x2 + 2x + 3? use either the remainder theorem or the factor theorem to explain your reasoning.

Since P(2) ≠ 0, hence (x-2) is not a factor of the polynomial

Given the polynomial function P(x) = x³ − 2x² + 2x + 3, we are to determine whether x - 2 is a factor.

According to the remainder theorem,

Let r(x) = x - 2

If r(x) = 0,

x - 2 = 0

x = 2

Substitute x = 2 into the polynomial function. If p(2) = 0, hence (x-2) is  factor of the polynomial.

P(x) = 2³ − 2(2)² + 2(6) + 3

P(2) = 8 - 8 + 6 + 3

P(2) = 9 ≠ 0

Since P(2) ≠ 0, hence (x-2) is not a factor of the polynomial

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We have , so by the remainder theorem is not a factor of . (The remainder would have been 0 if it were a factor.)