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Brazil algebra
Problem
The remainder on dividing the polynomial by (where and are unequal) is . Find the coefficients in terms of . Find for the case divided by and show that they are integers.
Solution
Let . So putting we get , . Solving, , .
In the case given , , so , . Note that , so and hence is a multiple of , so is an integer. , so is also an integer.
In the case given , , so , . Note that , so and hence is a multiple of , so is an integer. , so is also an integer.
Final answer
General: m = (p(a) − p(b)) / (a − b), n = (a p(b) − b p(a)) / (a − b). Specific (p(x) = x^200, divisor x^2 − x − 2): m = (2^200 − 1) / 3, n = (2^200 + 2) / 3, which are integers.
Techniques
Polynomial operationsModular Arithmetic