Solve the following problems and show your complete solutions. Write it on a paper and do not type your answer. 1.) Let V be the set of all polynomials of exactly degree 2 with the definitive of addition and scalarmultiplication as in example 6.EXAMPLE 6Another source of examples are sets of polynomials; therefore, we recall somewell-known facts about such functions. A polynomial (in 1) is a function that isexpressible asp(t) =ant"+an-1" + + a₁ + ao.where do. 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 highestpower of a term having a nonzero coefficient; p(1)=2r + 1 has degree 1, and theconstant polynomial p(t) = 3 has degree 0. The zero polynomial, denoted by 0.has no degree. We now let P₁, be the set of all polynomials of degree ≤n togetherwith the zero polynomial. If p(t) and q (t) are in P, we can writep(t) =ant" +an-1-++at+aoandq(t)=bat" +b-11 +...+bit + bo.We define (1) q(1) asp(t) @q(t) = (a + b)t + (an-1+ba-1)-1++ (a₁ + b₁)t + (ao+bo).If c is a scalar, we also define c o p(1) ascop(t) = (ca,)t" + (ca-1)++(ca)r + (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 ofdegreen or the zero polynomial. Then the previous definitions of the operationsand Ⓒ show that p(t)q(1) and cop(1), for any scalar e, are polynomials ofdegreen or the zero polynomial. That is, p(t)q(t) and eo p(t) are in P, sothat (a) and (b) in Definition 4.4 hold. To verify property (1), we observe thatg (1) p(1) = (b, + a)t" +(bn-1+ an-1)-1++ (b₁ + ₁)² + (ao+bo).and since a, + b₁ = b; +a, holds for the real numbers, we conclude that p(r)q(1) = q (1) p(1). Similarly, we verify property (2). The zero polynomial is theelement 0 needed in property (3). If p() is as given previously, then its negative,-p(t), is--ast-ag.We shall now verify property (6) and will leave the verification of the remainingproperties to the reader. Thus(c+d) Ⓒ p(t) = (c+d)ant" +(c+d)an-1-1++(c+d)at+ (c + d)ao= cant" + dant" + can-1"+dan-1-1+...+cart+ dayt + cao + dao= c(ant" | an-1"-+-+at+ap)+d(ant"+an-1t++at+ao)=cop(1) dop(1).a.) Show that V is not closes under addition

Answered: Image /qna-images/answer/e034d691-7cb0-4b94-b46b-25cca5f47dbd.jpg

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