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Difference between revisions of "2004 AMC 12A Problems/Problem 23"

m (→‎See also: typo fix)
(In the explanation of B, c_2003 isn't necessarily the sum of the roots.)
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*<math>\mathrm{(A)}</math>: We are given that <math>a_1 = b_1 = 0</math>, so <math>z_1 = 0</math>. If one of the roots is zero, then <math>P(0) = c_0 = 0</math>.  
 
*<math>\mathrm{(A)}</math>: We are given that <math>a_1 = b_1 = 0</math>, so <math>z_1 = 0</math>. If one of the roots is zero, then <math>P(0) = c_0 = 0</math>.  
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*<math>\mathrm{(B)}</math>: By [[Vieta's formulas]], we know that <math>c_{2003}</math> is the product of all of the roots of <math>P(x)</math>. But we know that <math>z_1 = 0</math>, so the product <math>c_{2003} = 0</math>.
+
*<math>\mathrm{(B)}</math>: By [[Vieta's formulas]], we know that <math>\frac{c_{2003}}{c_{2004}</math> is the sum of all of the roots of <math>P(x)</math>. Since that is real, <math> \sum_{k = 1}^{2004}{b_k}=0=\sum_{k = 1}^{2004}{a_k}, and </math>\frac{c_{2003}}{c_{2004}<math>, so </math>c_{2003}=0<math>.
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*<math>\mathrm{(C)}</math>: All of the coefficients are real. For sake of contradiction suppose none of <math>b_{2\ldots 2004}</math> are zero. Then for each complex root <math>z_i</math>, its [[complex conjugate]] <math>\overline{z_i} = a_i - b_ik</math> is also a root. So the roots should pair up, but we have an odd number of imaginary roots! This gives us the contradiction, and therefore the product is equal to zero.
+
*</math>\mathrm{(C)}<math>: All of the coefficients are real. For sake of contradiction suppose none of </math>b_{2\ldots 2004}<math> are zero. Then for each complex root </math>z_i<math>, its [[complex conjugate]] </math>\overline{z_i} = a_i - b_ik<math> is also a root. So the roots should pair up, but we have an odd number of imaginary roots! This gives us the contradiction, and therefore the product is equal to zero.
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*<math>\mathrm{(D)}</math>: We are given that <math>\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}</math>. Since the coefficients are real, it follows that if a root is complex, its conjugate is also a root; and the sum of the imaginary parts of complex conjugates is zero. Hence the RHS is zero.
+
*</math>\mathrm{(D)}<math>: We are given that </math>\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}<math>. Since the coefficients are real, it follows that if a root is complex, its conjugate is also a root; and the sum of the imaginary parts of complex conjugates is zero. Hence the RHS is zero.
  
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There is, however, no reason to believe that <math>\boxed{\mathrm{E}}</math> should be zero (in fact, that quantity is <math>P(1)</math>, and there is no evidence that <math>1</math> is a root of <math>P(x)</math>).  
+
There is, however, no reason to believe that </math>\boxed{\mathrm{E}}<math> should be zero (in fact, that quantity is </math>P(1)<math>, and there is no evidence that </math>1<math> is a root of </math>P(x)$).  
  
 
== See also ==
 
== See also ==

Revision as of 10:07, 4 December 2007

Problem

A polynomial

\[P(x) = c_{2004}x^{2004} + c_{2003}x^{2003} + ... + c_1x + c_0\]

has real coefficients with $c_{2004}\not = 0$ and $2004$ distinct complex zeroes $z_k = a_k + b_ki$, $1\leq k\leq 2004$ with $a_k$ and $b_k$ real, $a_1 = b_1 = 0$, and

\[\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}.\]

Which of the following quantities can be a nonzero number?

$\text {(A)} c_0 \qquad \text {(B)} c_{2003} \qquad \text {(C)} b_2b_3...b_{2004} \qquad \text {(D)} \sum_{k = 1}^{2004}{a_k} \qquad \text {(E)}\sum_{k = 1}^{2004}{c_k}$

Solution

We have to evaluate the answer choices and use process of elimination:

  • $\mathrm{(A)}$: We are given that $a_1 = b_1 = 0$, so $z_1 = 0$. If one of the roots is zero, then $P(0) = c_0 = 0$.
  • $\mathrm{(B)}$: By Vieta's formulas, we know that $\frac{c_{2003}}{c_{2004}$ (Error compiling LaTeX. Unknown error_msg) is the sum of all of the roots of $P(x)$. Since that is real, $\sum_{k = 1}^{2004}{b_k}=0=\sum_{k = 1}^{2004}{a_k}, and$\frac{c_{2003}}{c_{2004}$, so$c_{2003}=0$. *$\mathrm{(C)}$: All of the coefficients are real. For sake of contradiction suppose none of$b_{2\ldots 2004}$are zero. Then for each complex root$z_i$, its [[complex conjugate]]$\overline{z_i} = a_i - b_ik$is also a root. So the roots should pair up, but we have an odd number of imaginary roots! This gives us the contradiction, and therefore the product is equal to zero. *$\mathrm{(D)}$: We are given that$\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}$. Since the coefficients are real, it follows that if a root is complex, its conjugate is also a root; and the sum of the imaginary parts of complex conjugates is zero. Hence the RHS is zero.

There is, however, no reason to believe that$ (Error compiling LaTeX. Unknown error_msg)\boxed{\mathrm{E}}$should be zero (in fact, that quantity is$P(1)$, and there is no evidence that$1$is a root of$P(x)$).

See also

2004 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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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