Difference between revisions of "2002 AMC 12A Problems/Problem 25"
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− | ==Solution== | + | ==Solution 1== |
− | 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>.) |
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+ | ==Solution 2== | ||
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+ | We know every coefficient is equal, so we get <math>ax^n + ... + ax + a = 0</math> which equals <math>x^n + ... + x + 1 = 0</math>. We see apparently that x cannot be positive, for it would yield a number greater than zero for <math>Q(x)</math>. 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== | ==See Also== | ||
− | {{AMC12 box|year=2002|ab=A|num-b=24|after=Last | + | {{AMC12 box|year=2002|ab=A|num-b=24|after=Last Problem}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 01:25, 2 September 2024
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 1
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 |
Followed by Last Problem |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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