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MATH / Steve says to Jon, "I am thinking of a polynomial whose roots are all positive…

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problem

Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form P(x)=2x3−2ax2+(a2−81)x−cP(x) = 2x^3-2ax^2+(a^2-81)x-c for some positive integers aa and cc. Can you tell me the values of aa and cc?" After some calculations, Jon says, "There is more than one such polynomial." Steve says, "You're right. Here is the value of aa." He writes down a positive integer and asks, "Can you tell me the value of cc?" Jon says, "There are still two possible values of cc." Find the sum of the two possible values of cc.
Plain-text mathematical notation (without MathML)
Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form P(x)=2x³−2ax²+(a²−81)x−c for some positive integers a and c. Can you tell me the values of a and c?"

After some calculations, Jon says, "There is more than one such polynomial."

Steve says, "You're right. Here is the value of a." He writes down a positive integer and asks, "Can you tell me the value of c?"

Jon says, "There are still two possible values of c."

Find the sum of the two possible values of c.
Original LaTeX notation
Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $P(x) = 2x^3-2ax^2+(a^2-81)x-c$ for some positive integers $a$ and $c$. Can you tell me the values of $a$ and $c$?"

After some calculations, Jon says, "There is more than one such polynomial."

Steve says, "You're right. Here is the value of $a$." He writes down a positive integer and asks, "Can you tell me the value of $c$?"

Jon says, "There are still two possible values of $c$."

Find the sum of the two possible values of $c$.

level

Level 4

type

Intermediate Algebra

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Source and history

task_id
9573ff44-b80e-5372-81b1-568ae76a4f83
task_key
intermediate~5falgebra--test--9573ff44-b80e-5372-81b1-568ae76a4f83
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import