Difference between revisions of "2002 AMC 12A Problems/Problem 25"
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− | The sum of the coefficients of <math>P</math> and of <math>Q</math> will be equal, so <math>P(1) = Q(1)</math>. The only answer choice with an intersection between the two graphs at <math>x = 1</math> is | + | The sum of the coefficients of <math>P</math> and of <math>Q</math> will be equal, so <math>P(1) = Q(1)</math>. The only answer choice with an intersection between the two graphs at <math>x = 1</math> is '''(B)'''. (The polynomials in the graph are <math>P(x) = 2x^4-3x^2-3x-4</math> and <math>Q(x) = -2x^4-2x^2-2x-2</math>.) |
==Solution 2== | ==Solution 2== |
Revision as of 17:08, 17 January 2021
Contents
[hide]Problem
The nonzero coefficients of a polynomial with real coefficients are all replaced by their mean to form a polynomial . Which of the following could be a graph of and over the interval ?
Solution
The sum of the coefficients of and of will be equal, so . The only answer choice with an intersection between the two graphs at is (B). (The polynomials in the graph are and .)
Solution 2
We know every coefficient is equal, so we get which equals . We see apparently that x cannot be positive, for it would yield a number greater than zero for . We look at the zeros of the answer choices. A, C, D, and E have a positive zero, which eliminates them. B is the answer.
See Also
2002 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
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All AMC 12 Problems and Solutions |
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