5. Let n be a fixed positive integer. Denote by Pp the set of all real polynomials of degreen or less (including n = 0). Define addition and multiplication byP(x) O q(x) = (ao + a1x + a2x² + . . . + anx") + (bo + b1x + b2x² + + bnx“)3(ao+ bo) + (aı + b1)x + (a2 + b2)x² + · ··...+ (an + bn)x"..andcO p(x) = c· (ao + a1x + azx² + . . . + a,x") = cao + ca1x + ca2x +.+ canx"+.. ·+ anx") = cao + ca1x + ca2x + . .+ camx"Verify that V is a vector space.

A nonempty set V together with two operations - vector addition and scalar multiplication is called…

n 5. Let n be a fixed positive integer. Denote by Po the set of all real polynomials of degree n or less (including n = 0). Define addition and multiplication by p(x) q(x) = (ao + a1x + azx² + . . . + a,x") + (bo + b1x + b2x² + · ··+ bna") = (ao + bo) + (a1 + b1)x + (a2 + b2)x² + · . . + (an + bn)x" and co p(x) = cao + ca1x + ca2x² + . . .+ ca,x" = c. (ao + a1x + a2x2 + . . .+ anx") Verify thatV is a vector space.

n 5. Let n be a fixed positive integer. Denote by Po the set of all real polynomials of degree n or less (including n = 0). Define addition and multiplication by p(x) q(x) = (ao + a1x + azx² + . . . + a,x") + (bo + b1x + b2x² + · ··+ bna") = (ao + bo) + (a1 + b1)x + (a2 + b2)x² + · . . + (an + bn)x" and co p(x) = cao + ca1x + ca2x² + . . .+ ca,x" = c. (ao + a1x + a2x2 + . . .+ anx") Verify thatV is a vector space.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 13E

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