Question

1 point Consider the set P3 = (ar3 + br2 + cr + d | a,b,c,dER) of polynomials of degree at most 3 with real coefficients. Addition and scalar multiplication of polynomials are defined as usual, i.e., =(a+d)x3 + (b + b,)x2 + (c + c')x + (d + d'), k(ax3 + bx2 + cx + d) = kar3 +kbx2 +kx + kd. Our goal is to prove that P3, with these operations, is a vector space. In this problem, you are asked to prove axiom (SM1). Put 9 of the following sentence fragments in order to form a logically correct proof of (SM1). There is only one correct answer, so be sure not to skip any steps

Answer 1

Consider the set, P -ax' +bx* +cx + d|a,b,c, d eR From the data, addition and scalar multiplication of polynomials are defined as, (ax' +bx* +cx +d)+(a+b'x +cx+d) -(a+d)r(b+b)x2 +(c+c')x+(d+d') Let, , p(x)=ar +ax +ax +a, Axiom SM1 is, k(p(x)+(x)-kp(x)+k(x) Proof of SM1: Consider k(p(x)+q(x)) -k(a,+b)r+(a+b(a+h)x (a+b F =k(4 +ム)x3 +k(a2 + 4)x" + k(a +4)x+k(4+4) From definition (ii)] [From definition ] kp(x)+ kq(x) Thus k(p(x)+x)kp(x)+k(x) Hence proved axiom SM1 Thus, correct proof of SMl is provided above