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MATH / Consider all polynomials of a complex variable, P(z)=4z⁴+az³+bz²+cz+d, where a,b,c, and d are integers, 0≤d≤c≤b≤a≤4, and the polynomial has a zero z₀ with |z₀|=1. What is the sum o…

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problem

Consider all polynomials of a complex variable, P(z)=4z4+az3+bz2+cz+dP(z)=4z^4+az^3+bz^2+cz+d, where a,b,c,a,b,c, and dd are integers, 0≤d≤c≤b≤a≤40\le d\le c\le b\le a\le 4, and the polynomial has a zero z0z_0 with |z0|=1.|z_0|=1. What is the sum of all values P(1)P(1) over all the polynomials with these properties?
Plain-text mathematical notation (without MathML)
Consider all polynomials of a complex variable, P(z)=4z⁴+az³+bz²+cz+d, where a,b,c, and d are integers, 0≤d≤c≤b≤a≤4, and the polynomial has a zero z₀ with |z₀|=1. What is the sum of all values P(1) over all the polynomials with these properties?
Original LaTeX notation
Consider all polynomials of a complex variable, $P(z)=4z^4+az^3+bz^2+cz+d$, where $a,b,c,$ and $d$ are integers, $0\le d\le c\le b\le a\le 4$, and the polynomial has a zero $z_0$ with $|z_0|=1.$ What is the sum of all values $P(1)$ over all the polynomials with these properties?

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
7eeb9e54-c078-54a3-a44f-a30b182f6b30
task_key
intermediate~5falgebra--test--7eeb9e54-c078-54a3-a44f-a30b182f6b30
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import