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MATH / A polynomial P(x)=c₂₀₀₄x²⁰⁰⁴+c₂₀₀₃x²⁰⁰³+⋯+c₁x+c₀has real coefficients with c₂₀₀₄≠0 and 2004 distinct complex zeros z_(k)=a_(k)+b_(k)i, 1≤k≤2004 with a_(k) and b_(k) real, a₁=b₁=0, …

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A polynomial P(x)=c2004x2004+c2003x2003+⋯+c1x+c0 P(x)=c_{2004}x^{2004}+ c_{2003}x^{2003}+ \cdots+ c_{1}x+ c_{0} has real coefficients with c2004≠0c_{2004} \neq 0 and 2004 distinct complex zeros zk=ak+bkiz_{k}=a_{k}+ b_{k}i, 1≤k≤20041 \leq k \leq 2004 with aka_k and bkb_k real, a1=b1=0a_1 = b_1 = 0, and ∑k=12004ak=∑k=12004bk. \sum_{k=1}^{2004} a_{k}= \sum_{k=1}^{2004} b_{k}. Which of the following quantities can be a nonzero number? A. c0c_0 B. c2003c_{2003} C. $b_{2}b_{3} \dotsm b_{2004}$ D. ∑k=12004ak\sum_{k=1}^{2004}a_{k} E. ∑k=12004ck\sum_{k=1}^{2004}c_{k}
Plain-text mathematical notation (without MathML)
A polynomial P(x)=c₂₀₀₄x²⁰⁰⁴+c₂₀₀₃x²⁰⁰³+⋯+c₁x+c₀has real coefficients with c₂₀₀₄≠0 and 2004 distinct complex zeros z_(k)=a_(k)+b_(k)i, 1≤k≤2004 with a_(k) and b_(k) real, a₁=b₁=0, and ∑_(k=1)²⁰⁰⁴a_(k)=∑_(k=1)²⁰⁰⁴b_(k).Which of the following quantities can be a nonzero number?

A. c₀

B. c₂₀₀₃

C. $b_{2}b_{3} \dotsm b_{2004}$

D. ∑_(k=1)²⁰⁰⁴a_(k)

E. ∑_(k=1)²⁰⁰⁴c_(k)
Original LaTeX notation
A polynomial \[
P(x)=c_{2004}x^{2004}+ c_{2003}x^{2003}+ \cdots+ c_{1}x+ c_{0}
\]has real coefficients with $c_{2004} \neq 0$ and 2004 distinct complex zeros $z_{k}=a_{k}+ b_{k}i$, $1 \leq k \leq 2004$ 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?

A. $c_0$

B. $c_{2003}$

C. $b_{2}b_{3} \dotsm b_{2004}$

D. $\sum_{k=1}^{2004}a_{k}$

E. $\sum_{k=1}^{2004}c_{k}$

level

Level 4

type

Intermediate Algebra

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Source and history

task_id
93a34c6c-05de-594b-b3b1-876731564506
task_key
intermediate~5falgebra--test--93a34c6c-05de-594b-b3b1-876731564506
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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