Let P_{2}denote the set of all polynomials of degree 2 or less.That is, P_{2} is the set of all functions of the form
p(x) = a_{2}x^{2} +a_{1}x +a_{0}
where a_{0},a_{1},a_{2} ∈ R. Show that P_{2}is closed under addition and scalar multiplication, and that P_{2}contains a zero element.
Before we formally address the stated tasks, let us remind ourselves how we add polynomial functions. If we are given, say, f (x)=2x_{2}−5x+11 and g (x)=4x −3, we compute (f +g )(x)=f (x)+g (x)=2x^{2}−5x +11+4x −3. We can then add like terms to simplify and find that (f +g )(x) = 2x^{2} −x +8.
Similarly, if we wanted to compute (−3f )(x), we have (−3f )(x)=−3f (x) =−3(2x^{2} −5x +11)=−6x^{2} +15x −33.
We now show that P_{2} is indeed closed under the operations of addition and scalar multiplication. Given two arbitrary elements of P_{2}, say f (x)=a_{2}x^{2}+a_{1}x +a_{0} and g(x) = b_{2}x^{2} +b_{1}x +b_{0}, it follows upon adding and combining like terms that
(f +g )(x) = (a_{2} +b_{2})x^{2} +(a_{1} +b_{1})x +(a_{0} +b_{0})
which is obviously a polynomial of degree 2 or lower, and thus f + g is an element of P_{2}. In the same way, for any real value c,
(cf )(x) = ca_{2}x^{2} +ca{1}x +ca_{0}
which also belongs to P_{2}. Finally, it is evident that if we let z(x) = 0x^{2} +0x +0 (i.e., z(x) is the zero function), then (f +z)(x)=f (x) for any choice of f in P_{2}.