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A cubic polynomial P(x) has leading coefficient 3 and satisfies P(1) = 1, P(2) = 2, P(3) = 3. Find P(4).

1. 4
2. 22
3. 28
4. 7

Correct answer: 22

Solution

Let Q(x) = P(x) - x. Since P(1)=1, P(2)=2, P(3)=3, Q has roots 1, 2, 3. P is cubic with leading coefficient 3, and subtracting x (degree 1) does not change the leading term, so Q(x) = 3*(x-1)*(x-2)*(x-3). Then P(x) = 3*(x-1)*(x-2)*(x-3) + x, so P(4) = 3*(3)*(2)*(1) + 4 = 18 + 4 = 22.

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