Answer :

Given, x^{3} – 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 x^{2} + ax + b

∴ p(x) = (x – a)( x^{2} + ax + b) + c

⇒ x^{3} – 1 = (x + 1)( x^{2} + ax + b) + c

⇒ x^{3} – 1 = (x^{3} + x^{2} + ax^{2} + ax + bx + b) + c

⇒ x^{3} – 1 = x^{3} + (a + 1)x^{2} + (a + b)x + (c + b)

∴ a + 1 = 0, a + b = 0, c + b = – 1

⇒ a = – 1

⇒ a + b = 0

⇒ – 1 + b = 0

⇒ b = 1

⇒ c + b = – 1

⇒ c + 1 = – 1

⇒ c = – 2

Quotient = x^{2} + ax + b = x^{2} – x + 1

Remainder = – 2

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