2004 AMC 12A Problems/Problem 23
Problem
has real coefficients with and
distinct complex zeroes
,
with
and
real,
, and
Which of the following quantities can be a nonzero number?
Solution
We have to evaluate the answer choices and use process of elimination:
: We are given that
, so
. If one of the roots is zero, then
.
: 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
. Since that is real,
\frac{c_{2003}}{c_{2004}
c_{2003}=0
\mathrm{(C)}
b_{2\ldots 2004}
z_i
\overline{z_i} = a_i - b_ik
\mathrm{(D)}
\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}}P(1)
1
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 |