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Let p(x) be a cubic polynomial with leading coefficient 1 satisfying p(1) = 1, p(2) = 4, and p(3) = 9. Which of the following statements is/are correct?

1. p(4) = 22
2. p(6/5) = (6/5)²
3. p(x) = x² holds for exactly two values of x
4. p(x) = 0 has a root in the interval (0, 1)

Correct answer: p(4) = 22

Solution

Setting g(x) = p(x) - x² gives a monic cubic vanishing at x = 1, 2, 3, so g(x) = (x-1)(x-2)(x-3). All statements can be checked from this explicit form.

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