Difference between revisions of "2007 AMC 12A Problems/Problem 18"
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The [[polynomial]] <math>f(x) = x^{4} + ax^{3} + bx^{2} + cx + d</math> has real [[coefficient]]s, and <math>f(2i) = f(2 + i) = 0.</math> What is <math>a + b + c + d?</math> | The [[polynomial]] <math>f(x) = x^{4} + ax^{3} + bx^{2} + cx + d</math> has real [[coefficient]]s, and <math>f(2i) = f(2 + i) = 0.</math> What is <math>a + b + c + d?</math> | ||
Revision as of 08:41, 14 March 2010
Problem
The polynomial 
has real coefficients, and 
What is 
Solution
A fourth degree polynomial has four roots. Since the coefficients are real, the remaining two roots must be the complex conjugates of the two given roots, namely 
. Now we work backwards for the polynomial:
Thus our answer is 
.
See also
| 2007 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 17 |
Followed by Problem 19 |
| 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 | |
