Is the set of all polynomials in #P2# of the form #a_0 + a_1 x + a_2 x^2# where #a_0 = a_2 ^2# closed under addition?

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Answer 1 · 2017-04-01T20:10:55

See below.

If

#p_a(x)=a_2x^2+a_1x+a_2^2# and
#p_b(x)=b_2x^2+b_1x+b_2^2#

#p_a(x)+p_b(x) = (a_2+b_2)x^2+(a_1+b_1)x+a_2^2+b_2^2#

As we can observe, to be closed under addition we must have

#p_a(x)+p_b(x) = (a_2+b_2)x^2+(a_1+b_1)x+(a_2+b_2)^2#

but #(a_2+b_2)^2 ne a_2^2+b_2^2# so polynomials with the strucure

#p_a(x)=a_2x^2+a_1x+a_2^2# are not closed under addition.