Answer :

Given, a pair of polynomial as x – 1, x3 + 4x2 – 3x – 6


Need to find out the first polynomial is factor of second and if not a factor need to find the remainder


To check x – 1 is a factor of x3 + 4x2 – 3x – 6 we must substitute x = 1 in the second polynomial, we get as follows


1 + 4 – 3 – 6 = – 4 not equal to 0


x – 1 is not a factor of x3 + 4x2 – 3x – 6


To find the remainder by using divide second polynomial by first polynomial


so, we can subtract a number from the second polynomial to get the remainder


x3 + 4x2 – 3x – 6 = (x – 1)q(x) + c


x3 + 4x2 – 3x – 6 –c = (x – 1)q(x)


c = ((x3 + 4x2 – 3x – 6) – (x – 1)) × q(x)


Now, substitute x = 1 in the above equation we get


c = (1 + 4 – 3 – 6 – 1 + 1) × q(1)


c = – 4


– 4 is the remainder


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