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Difference between revisions of "2007 AMC 12A Problems/Problem 18"

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==Solution==
 
==Solution==
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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:
+
A fourth degree polynomial has four [[root]]s. Since the coefficients are real(meaning that complex roots come in conjugate pairs), 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 />
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<math>x^4 - 4x^3 + 9x^2 - 16x + 20 = 0</math></div>
 
<math>x^4 - 4x^3 + 9x^2 - 16x + 20 = 0</math></div>
  
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Thus our answer is <math>- 4 + 9 - 16 + 20 = 9\ \mathrm{(D)}</math>.  
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Thus our answer is <math>- 4 + 9 - 16 + 20 = 9\ \mathrm{(D)}</math>.
  
 
==See also==
 
==See also==

Revision as of 19:43, 27 October 2013

Problem

The polynomial $f(x) = x^{4} + ax^{3} + bx^{2} + cx + d$ has real coefficients, and $f(2i) = f(2 + i) = 0.$ What is $a + b + c + d?$

$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ 4 \qquad \mathrm{(D)}\ 9 \qquad \mathrm{(E)}\ 16$

Solution

A fourth degree polynomial has four roots. Since the coefficients are real(meaning that complex roots come in conjugate pairs), the remaining two roots must be the complex conjugates of the two given roots, namely $2-i,-2i$. Now we work backwards for the polynomial:

$(x-(2+i))(x-(2-i))(x-2i)(x+2i) = 0$

$(x^2 - 4x + 5)(x^2 + 4) = 0$

$x^4 - 4x^3 + 9x^2 - 16x + 20 = 0$

Thus our answer is $- 4 + 9 - 16 + 20 = 9\ \mathrm{(D)}$.

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

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