(i) p(x) = 2x2 - 13x + 13, g(x) = 3 + x
∵ 2x2 - 13x + 13 = (2x - 19) (3 + x) + 70
Remainder theorem is
p(x) = q(x) g(x) + r(x)
∴ q(x) = 2x - 19
r(x) = 70
(ii) p(x) = x3 - x2 + 4x - 8, g(x) = x + 3
∵ x3 - x2 + 4x - 8 = (x2 - 4x + 16) (x + 3) - 56
∴ q(x) = x2 - 4x + 16
r(x) = -56
(iii) p(x) = 2x3 + 6x2 - x - 1, g(x) = x2 - 3
∵ 2x3 + 6x2 - x - 1 = (2x + 6) (x2 - 3) + 5x + 17
∴ q(x) = 2x + 6
r(x) = 5x + 17
(iv) p(x) = 3x3 + 8x2 + 4, g(x) = 7 - x2
∵ 3x3 + 8x2 + 4 = (- 3x - 8) (7 - x2) + 21x + 6
\(\)∴ q(x) = -3x - 8 = - (3x + 8)
r(x) = 21x + 6
(v) p(x) = x3 + 12x2 + 7x, g(x) = 3 - 2x - x2
∵ x3 + 12x2 + 7x = (-x - 10) (3 - 2x - x2) -10x + 30
\(\)∴ q(x) = -x - 10 = -(x + 10)
r(x) = -10x + 30