HMMT February

Algebra Difficulty 4.9 AIME Find the answer

P is a polynomial. When P is divided by $x-1$ , the remainder is -4 . When P is divided by $x-2$ , the remainder is -1 . When $P$ is divided by $x-3$ , the remainder is 4 . Determine the remainder when $P$ is divided by $x^{3}-6 x^{2}+11 x-6$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The remainder polynomial is simply the order two polynomial that goes through the points $(1,-4),(2,-1)$ , and $(3,4): x^{2}-5$ .

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