For each pair of polynomial is below, check whether the first is a factor of the second. If not a factor, find the remainder on dividing the second by the first.

x – 1, x³ + 4x² – 3x – 6

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