1.) Let V be the set of all polynomials of exactly degree 2 with the definitive of addition and scalar multiplication as in example 6. EXAMPLE 6 Another source of examples are sets of polynomials; therefore, we recall some well-known facts about such functions. A polynomial (in ) is a function that is expressible as p(t) =ant"+an-1" + + a₁ + ao. where ao. a₁.....a, are real numbers and n is a nonnegative integer. If a,, # 0. then p(t) is said to have degree n. Thus the degree of a polynomial is the highest power of a term having a nonzero coefficient; p(1)=2r + 1 has degree 1, and the constant polynomial p(t) = 3 has degree 0. The zero polynomial, denoted by 0. has no degree. We now let Pr, be the set of all polynomials of degreen together with the zero polynomial. If p(t) and q (t) are in P, we can write p(t) =ant" +an-1-¹++ a₁t+ao and q(t)=bat" +bn-1t+...+bit + bo. We define p(1)q(t) as p(t) @q(t) = (a + b)!" + (an-1+ba-1)-1++ (a₁ + b₁)t + (ao+bo). If c is a scalar, we also define c o p(1) as cop(t) = (ca,)t" + (ca,-1)"++(ca₁)t + (cap). We now show that P, is a vector space. Let p(t) and q (1), as before, be elements of P₁; that is, they are polynomials of degreen or the zero polynomial. Then the previous definitions of the operations and Ⓒ show that p(t)q(r) and ep(1), for any scalar e, are polynomials of degreen or the zero polynomial. That is, p(t)q(t) and c o p(t) are in P, so that (a) and (b) in Definition 4.4 hold. To verify property (1), we observe that q(1) p(t) = (b +a,)t" + (bn-1+an-1) ++ (b₁ +₁)r+ (ao + bo). and since a, + b, = b; +a, holds for the real numbers, we conclude that p(1) q(1) = q(1) p (1). Similarly, we verify property (2). The zero polynomial is the element 0 needed in property (3). If p() is as given previously, then its negative, -p(t), is astag. We shall now verify property (6) and will leave the verification of the remaining properties to the reader. Thus (c+d) Ⓒ p(t) = (c + d)ant" + (c+d)an-it-1+...+(c+d)at + (c +d)ao = cant" + dant" + can-1"+dan-1-1+...+cart +dayt + cao + dao =c(at" a-t-+ + a₁ + a) + d(ant"+an-1"++at+ao) =cop(1) @dop(1). a.) Show that V is not closes under addition