Using remainder theorem, find the remainder on dividing f(x) by (x + 3) where

(i) f(x) = 2x² - 5x + 1

(ii) f(x) = 3x³ + 7x² - 5x + 1

(i) By remainder theorem, on dividing f(x) by (x - a) , remainder = f(a)

∴ On dividing, f(x) = 2x² - 5x + 1 by (x + 3) or (x - (-3))

Remainder = f(-3)

$= 2(-3)^2 - 5(-3) + 1 \\[0.5em] = 2(9) + 15 + 1 \\[0.5em] = 18 + 16 \\[0.5em] = 34.$

Hence, the value of remainder is 34.

(ii) By remainder theorem, on dividing f(x) by (x - a) , remainder = f(a)

∴ On dividing, f(x) = 3x³ + 7x² - 5x + 1 by (x + 3) or (x - (-3))

Remainder = f(-3)

$= 3(-3)^3 + 7(-3)^2 - 5(-3) + 1 \\[0.5em] = -81 + 63 + 15 + 1 \\[0.5em] = -2. \\[0.5em]$

Hence, the value of remainder is -2.