POLYNOMIALS AND THEIR ZEROS AMC
Polynomials 1999 AHSME #17
Let
be a polynomial which when divided by x - 19 has the remainder 99, and when divided by x – 99 has remainder 19. What is the remainder when 
is divided by (x – 19)(x – 99)?
Let
be a polynomial which when divided by x - 19 has the remainder 99, and when divided by x – 99 has remainder 19. What is the remainder when is divided by (x – 19)(x – 99)? OLMYPIAD >
POLYNOMIALS AND THEIR ZEROS >
1999 AHSME #17
May want to check out:
2000 AMC10 #24
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POLYNOMIALS AND THEIR ZEROS >
1999 AHSME #17
May want to check out:
2000 AMC10 #24
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VIDEO LECTURE
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SOLUTION
This question can look intimidating on first sight but after some careful implementation of the various theorems, the answer pretty much falls in place. We notice that (x – 19)(x – 99) is a quadratic polynomial. The remainder when this is divided into
for some constants a and b. If we were to put
Subtracting these equations an substituting gives
So a = -1 and b = 99 - (-1)19 =118. The remainder is therefore – x + 118. All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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will be linear, that is,
and 
, the linear factor theorem implies that
