(i) f(x) = x3 - 6x2 + 11x - 6 and g(x) = x2 + x + 1
Degree of f(x) is 3 and degree of g(x) is 2;therefore degree of q(x) is 3 - 2 = 1 and degree of remainder is less than 2,
Let q(x) = ax + b and r(x) = cx + d
By applying division algorithm:
Dividend = Quotient× Divisor + Remainder
f(x) = qx x g(x) + r(x)
On substituting values in the above relation we get,
On comparing coefficients we get,
a = 1
a + b = - 6
a + b + c = 11
b + d = - 6
On solving above equations we get,
a = 1,b = - 7,c = 17, d = 1
On substituting these values for q(x) and r(x)
(ii) f(x) = 10x4 + 17x3 - 62x2 + 30x - 3 and g(x) = 2x2 + 7x + 1
Degree of f(x) = 10x4 + 17x3 - 62x2 + 30x - 3 ; therefore degree of q(x) is 4 - 2 = 2 and degree of remainder is less than 2.
Let q(x) = ax2 + bx + c and r(x) = px + 2
By applying division algorithm:
Dividend = Quotient× Divisor + Remainder
f(x) = q(x) x g(x) + r(x)
On substituting values in the above relation we get,
On comparing coefficients we get,
On solving above equations we get,
a = 1, b= -9, c = -2,p = 53; q c= -1
On substituting these values for q(x) and r(x)