Answer :

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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