# 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?

Apr 1, 2017

See below.

#### Explanation:

If

${p}_{a} \left(x\right) = {a}_{2} {x}^{2} + {a}_{1} x + {a}_{2}^{2}$ and
${p}_{b} \left(x\right) = {b}_{2} {x}^{2} + {b}_{1} x + {b}_{2}^{2}$

${p}_{a} \left(x\right) + {p}_{b} \left(x\right) = \left({a}_{2} + {b}_{2}\right) {x}^{2} + \left({a}_{1} + {b}_{1}\right) x + {a}_{2}^{2} + {b}_{2}^{2}$

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

${p}_{a} \left(x\right) + {p}_{b} \left(x\right) = \left({a}_{2} + {b}_{2}\right) {x}^{2} + \left({a}_{1} + {b}_{1}\right) x + {\left({a}_{2} + {b}_{2}\right)}^{2}$

but ${\left({a}_{2} + {b}_{2}\right)}^{2} \ne {a}_{2}^{2} + {b}_{2}^{2}$ so polynomials with the strucure

${p}_{a} \left(x\right) = {a}_{2} {x}^{2} + {a}_{1} x + {a}_{2}^{2}$ are not closed under addition.