SupposeÂ Â is a polynomial with real coefficients, satisfying the conditionÂ ,Â for every realÂ .Â Prove thatÂ Â can be expressed in the formfor some real numbersÂ Â and non-negative integerÂ .

Using a very standard trigometric identity , we can easily convert the following ,

Assuming , Â for all realsÂ .Â So,

for allÂ .Â SinceÂ Â holds for infinitely manyÂ ,Â it must hold for allÂ Â (sinceÂ Â is a polynomial). so we get that ,Â Â Â isÂ a even polynomial .

Also

Â implies that

putting ,Â

for infinitely manyÂ Â .

so we get ,Â

Again as it is a polynomial function we can extend it allÂ .Â And we get ,Â Â for all realsÂ

Â implies that

putting ,Â

for infinitely manyÂ Â .

so we get ,Â

Again as it is a polynomial function we can extend it allÂ .Â And we get ,Â Â for all realsÂ

SinceÂ Â is even , we can choose a even polynomialÂ Â such thatÂ ,. now take ,Â Â and you get the polynomial of required form .

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