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MATH / There are nonzero integers a, b, r, and s such that the complex number r+si is a zero of the polynomial P(x)=x³−ax²+bx−65. For each possible combination of a and b, let p_(a,b) be …

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problem

There are nonzero integers aa, bb, rr, and ss such that the complex number r+sir+si is a zero of the polynomial P(x)=x3−ax2+bx−65P(x)={x}^{3}-a{x}^{2}+bx-65. For each possible combination of aa and bb, let pa,b{p}_{a,b} be the sum of the zeros of P(x)P(x). Find the sum of the pa,b{p}_{a,b}'s for all possible combinations of aa and bb.
Plain-text mathematical notation (without MathML)
There are nonzero integers a, b, r, and s such that the complex number r+si is a zero of the polynomial P(x)=x³−ax²+bx−65. For each possible combination of a and b, let p_(a,b) be the sum of the zeros of P(x). Find the sum of the p_(a,b)'s for all possible combinations of a and b.
Original LaTeX notation
There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x)={x}^{3}-a{x}^{2}+bx-65$. For each possible combination of $a$ and $b$, let ${p}_{a,b}$ be the sum of the zeros of $P(x)$. Find the sum of the ${p}_{a,b}$'s for all possible combinations of $a$ and $b$.

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
9a772e44-d961-5632-a678-fa998028465e
task_key
intermediate~5falgebra--test--9a772e44-d961-5632-a678-fa998028465e
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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