Difference between revisions of "2003 AMC 12B Problems/Problem 20"
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Two of the roots of <math>f(x) = 0</math> are <math>\pm 1</math>, and we let the third one be <math>n</math>. Then | Two of the roots of <math>f(x) = 0</math> are <math>\pm 1</math>, and we let the third one be <math>n</math>. Then | ||
<cmath>a(x-1)(x+1)(x-n) = ax^3-anx^2-ax+an = ax^3 + bx^2 + cx + d = 0</cmath> | <cmath>a(x-1)(x+1)(x-n) = ax^3-anx^2-ax+an = ax^3 + bx^2 + cx + d = 0</cmath> | ||
− | Notice that <math>f(0) = d = an = 2</math>, so <math>b = -an = -2</math>. | + | Notice that <math>f(0) = d = an = 2</math>, so <math>b = -an = -2 \Rightarrow \mathrm{(B)}</math>. |
== See also == | == See also == |
Revision as of 18:12, 7 April 2014
Problem
Part of the graph of is shown. What is ?
Solution
Solution 1
Since
It follows that . Also, , so .
Solution 2
Two of the roots of are , and we let the third one be . Then Notice that , so .
See also
2003 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
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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