Difference between revisions of "1987 AHSME Problems/Problem 24"
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\textbf{(E)}\ \infty </math> | \textbf{(E)}\ \infty </math> | ||
− | + | == Solution == | |
+ | Let <math>f(x) = \sum_{k=0}^{n} a_{k} x^{k}</math> be a polynomial satisfying the condition, so substituting it in, we find that the highest powers in each of the three expressions are, respectively, <math>a_{n}x^{2n}</math>, <math>a_{n}^{2}x^{2n}</math>, and <math>a_{n}^{n+1}x^{n^{2}}</math>. If polynomials are identically equal, each term must be equal, so we get <math>2n = n^2</math> and <math>a_{n} = a_{n}^{2} = a_{n}^{n+1}</math>, so since <math>n \geq 1</math>, we must have <math>n = 2</math>, and since <math>a_{n} \neq 0</math>, we have <math>a_{n} = 1</math>. The given condition now becomes <math>x^4 + bx^2 + c \equiv (x^2 + bx + c)^2</math>, so we must have <math>b = 0</math>, or else the right-hand side would have a cubic term that the left-hand side does not. Thus we get <math>x^4 + c \equiv (x^2 + c)^2</math>, so we must have <math>c = 0</math>, or else the right-hand side would have an <math>x^2</math> term that the left-hand side does not. Thus the only possibility is <math>f(x) = x^2</math>, i.e. there is only <math>1</math> solution, so the answer is <math>\boxed{\text{B}}</math>. | ||
== See also == | == See also == |
Latest revision as of 14:46, 1 March 2018
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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All AHSME Problems and Solutions |
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