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MATH / Let P(x)=24x²⁴+∑_(j=1)²³(24−j)(x^(24−j)+x^(24+j)).Let z₁,z₂,…,z_(r) be the distinct zeros of P(x), and let z_(k)²=a_(k)+b_(k)i for k=1,2,…,r, where i=√(−1), and a_(k) and b_(k) are…

Problem

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problem

Let P(x)=24x24+∑j=123(24−j)(x24−j+x24+j).P(x)=24x^{24}+\sum_{j=1}^{23}(24-j)\left(x^{24-j}+x^{24+j}\right).Let z1,z2,…,zrz_1, z_2, \ldots, z_r be the distinct zeros of P(x)P(x), and let zk2=ak+bkiz_k^2=a_k+b_{k}i for k=1,2,…,rk=1, 2, \ldots, r, where i=−1i=\sqrt{-1}, and aka_k and bkb_k are real numbers. Find ∑k=1r|bk|.\sum_{k=1}^{r}|b_k|.
Plain-text mathematical notation (without MathML)
Let P(x)=24x²⁴+∑_(j=1)²³(24−j)(x^(24−j)+x^(24+j)).Let z₁,z₂,…,z_(r) be the distinct zeros of P(x), and let z_(k)²=a_(k)+b_(k)i for k=1,2,…,r, where i=√(−1), and a_(k) and b_(k) are real numbers. Find
∑_(k=1)^(r)|b_(k)|.
Original LaTeX notation
Let $$P(x)=24x^{24}+\sum_{j=1}^{23}(24-j)\left(x^{24-j}+x^{24+j}\right).$$Let $z_1, z_2, \ldots, z_r$ be the distinct zeros of $P(x)$, and let $z_k^2=a_k+b_{k}i$ for $k=1, 2, \ldots, r$, where $i=\sqrt{-1}$, and $a_k$ and $b_k$ are real numbers. Find
\[\sum_{k=1}^{r}|b_k|.\]

level

Level 5

type

Precalculus

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Source and history

task_id
0f51e674-0d77-55f0-9317-7b491bdd8cef
task_key
precalculus--test--0f51e674-0d77-55f0-9317-7b491bdd8cef
task_revision_id
1
config / split
precalculus / test

Official source

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