L Find the remainder when r' + 3r + 3x + 1 is •divided by(i) % + 1 (ii) % - ! (iii) %2(iv) x + n (v) 6 + 2x.

Given: Let p(x) = x3 + 3x2 + 3x + 1. (i) When p(.x) is divided by x + 1, the remainder is p(- 1). .. p(- 1) = (- 1>3 + 3(- 1>2 + 3(- 1) + 1 = - 1 + 3 - 3 + 1= 0. (ii) When p(x ) is divided by (x - } the remairider is p ( ). :. P(i) = (H + a(i) + a(i) + 1 = 1 + 3 + 3 + l 8 4 2 1+6+12+8 'J:1 = 8 = (iii) When p(x) is divided by x, the remainder is p(O). :. p(O) = 0 + 0 + 0 + 1 = 1. (iv) When p( x) is divided by x + 1t, the remainder is p(- 1t). :. p(- x) = (- nJd + 3(- n>2 + 3(- x) + 1 = - r + an2 - aTt + 1. {v) When p(x ) is divided by 5 + 2%, the remainder is p(i) :. P(-i) = (-iJ + a(-iJ + a(-i) + 1 = - 125 + 75 15 - - + 1 8 4 2 -125+150-60 +8 -27 = 8 = 5•