1 point Consider the set P3 = (ar3 + br2 + cr + d | a,b,c,dER) of polynomials of

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

Explanation / Answer

We have k(p(x)+q(x)) = k( a3x3 +a2 x2 +a1 x +a0)+( b3x3 +b2 x2 +b1 x +b0 ))= ka3x3+ka2x2 + ka1x +ka0+ kb3x3+kb2x2+ kb1x +kb0 = k(a3 x3+ a2 x2+a1 x +a0)+k(b3 x3+ b2 x2+b1 x +b0)= kp(x)+kq(x).

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