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Omni-MATH / Determine the polynomials P of two variables so that:

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problem

Determine the polynomials P of two variables so that: [b]a.)[/b] for any real numbers t,x,yt,x,y we have P(tx,ty)=tnP(x,y)P(tx,ty) = t^n P(x,y) where nn is a positive integer, the same for all t,x,y;t,x,y; [b]b.)[/b] for any real numbers a,b,ca,b,c we have P(a+b,c)+P(b+c,a)+P(c+a,b)=0;P(a + b,c) + P(b + c,a) + P(c + a,b) = 0; [b]c.)[/b] P(1,0)=1.P(1,0) =1.
Plain-text mathematical notation (without MathML)
Determine the polynomials P of two variables so that:

[b]a.)[/b] for any real numbers t,x,y we have P(tx,ty)=t^(n)P(x,y) where n is a positive integer, the same for all t,x,y;

[b]b.)[/b] for any real numbers a,b,c we have P(a+b,c)+P(b+c,a)+P(c+a,b)=0;

[b]c.)[/b] P(1,0)=1.
Original LaTeX notation
Determine the polynomials P of two variables so that:

[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$

[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$

[b]c.)[/b] $P(1,0) =1.$

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Source and history

task_id
8402c309-21ec-5647-865a-335917a68b9f
task_key
test--8402c309-21ec-5647-865a-335917a68b9f
task_revision_id
1
config / split
/ test

Official source

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