2008 iTest Problems/Problem 80
Problem
Let
and let be the polynomial remainder when is divided by . Find the remainder when is divided by .
Solutions
Solution 1
. We apply the polynomial generalization of the Chinese Remainder Theorem.
Indeed,
since . Also,
using similar reasoning. Hence , and by CRT we have .
Then .
Solution 2
The given information can be represented as where is the remainder and is the quotient. Multiplying both sides by would make computation easier; doing so results in Now plug in values of not equal to 1 that make . By the Zero Product Property, the values of that make the expression equal to 0 are and .
By plugging in and into the equation and solving for the remainder function, we have . Since the remainder function's degree is at most 3 and we know four points, we can construct the unique remainder function.
Let . Plugging in results in , and plugging in results in . Thus, and . Plugging in results in . Plugging in results in . Thus, and , so
Substituting results in and . Therefore, , so , and the remainder when divided by 1000 is .
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 79 |
Followed by: Problem 81 | |
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