What are polynomials which f(x+y) = f(x) + f(y) + 2xy - 1 ?

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f(x)= x^2 +1

f(y)= y^2 + 1

==> Then we will find f(x+y).

f(x+y)= (x+y)^2 + 1

Now we will expand the brackets.

==> f(x+y)= x^2 + 2xy+ y^2 + 1

We will add and subtract 1 .

==> f(x+y)= x^2 + y^2 + 2xy + 1 + 1 -1

...

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f(x)= x^2 +1

f(y)= y^2 + 1

==> Then we will find f(x+y).

f(x+y)= (x+y)^2 + 1

Now we will expand the brackets.

==> f(x+y)= x^2 + 2xy+ y^2 + 1

We will add and subtract 1 .

==> f(x+y)= x^2 + y^2 + 2xy + 1 + 1 -1

Rearrange terms.

==> f(x+y)= (x^2 +1) + (y^2 + 1) + 2xy -1

==> f(x+y) =  f(x) + f(y) + 2xy -1

==> f(x)= x^2 + 1

==> f(y)= y^2 +1

 

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