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A polynomial P(x) leaves remainder 3 on division by (x - 1),…
Question
A polynomial P(x) leaves remainder 3 on division by (x - 1), remainder 5 on division by (x - 2), and remainder 7 on division by (x - 3). Let R(x) be the remainder when P(x) is divided by (x - 1)(x - 2)(x - 3). Compute R(10).
✓ Verified answer: 21checked by our engine — not a guess
Step-by-step solution
By the Remainder Theorem, P(1)=3, P(2)=5, P(3)=7.
The remainder on division by the degree-3 product is a polynomial R(x) of degree at most 2 with R(1)=3, R(2)=5, R(3)=7.
Since the points (1,3),(2,5),(3,7) are collinear on the line y = 2x + 1, the unique interpolating polynomial of degree ≤ 2 is exactly R(x) = 2x + 1 (its quadratic coefficient is 0).
Thus R(10) = 2·10 + 1 = 21.
Final answer21
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