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> | ||
| − | <math> | + | <math>\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ 4 \qquad \mathrm{(D)}\ 9 \qquad \mathrm{(E)}\ 16</math> |
==Solution== | ==Solution== | ||
| − | A fourth degree polynomial has four [[root]]s. Since the coefficients are real, the remaining two roots must be the [[complex conjugate]]s of the two given roots, namely <math>2+i, | + | A fourth degree polynomial has four [[root]]s. Since the coefficients are real, the remaining two roots must be the [[complex conjugate]]s of the two given roots, namely <math>2+i,-2i</math>. Now we work backwards for the polynomial: |
<div style="text-align:center;"><math>(x-(2+i))(x-(2-i))(x-2i)(x+2i) = 0</math><br /> | <div style="text-align:center;"><math>(x-(2+i))(x-(2-i))(x-2i)(x+2i) = 0</math><br /> | ||
<math>(x^2 - 4x + 5)(x^2 + 4) = 0</math><br /> | <math>(x^2 - 4x + 5)(x^2 + 4) = 0</math><br /> | ||
| − | <math> | + | <math>x^4 - 4x^3 + 9x^2 - 16x + 20 = 0</math></div> |
Thus our answer is <math>- 4 + 9 - 16 + 20 = 9\ \mathrm{(D)}</math>. | Thus our answer is <math>- 4 + 9 - 16 + 20 = 9\ \mathrm{(D)}</math>. | ||
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[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
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Revision as of 21:09, 30 November 2007
Solution
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 | |
