For each pair of polynomials below, find the quotient and remainder on dividing the first by the second.

x³ + 1, x – 1

Given, x³ – 1, x – 1 as pair of polynomials

Need to find the quotient and remainder

⇒ To find the quotient and remainder the given equation can be written as p(x) = (x – a)q(x) + b

⇒ Since, the polynomial is of third degree we can write the q(x) as x² + ax + b

∴ p(x) = (x – a)( x² + ax + b) + c

⇒ x³ + 1 = (x – 1)(x² + ax + b) + c

⇒ x³ + 1 = (x³ –x² + ax² – ax + bx – b) + c

⇒ x³ + 1 = x³ + (a–1)x² + (b – a)x + (c – b)

∴ a – 1 = 0, b – a = 0, c – b = 1

⇒ a = 1, b = 1

⇒ c – b = 1

⇒ c = 2

Quotient = x² + ax + b = x² + x + 1

Remainder = 2