Q2 of 31 Page 245

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 + (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 = x2 + ax + b = x2 – x + 1


Remainder = – 2


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