1987 AHSME Problems/Problem 24
Problem
How many polynomial functions of degree satisfy ?
Solution
Let be a polynomial satisfying the condition, so substituting it in, we find that the highest powers in each of the three expressions are, respectively, , , and . If polynomials are identically equal, each term must be equal, so we get and , so since , we must have , and since , we have . The given condition now becomes , so we must have , or else the right-hand side would have a cubic term that the left-hand side does not. Thus we get , so we must have , or else the right-hand side would have an term that the left-hand side does not. Thus the only possibility is , i.e. there is only solution, so the answer is .
See also
1987 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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