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.

3x – 1, 3x³ – 10x² + 9x – 2

Given, a pair of polynomial as 3x – 1, 3x³ – 10x² + 9x – 2

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

⇒ To check 3x – 1 is a factor of 3x³ – 10x² + 9x – 2 we must substitute x = in the second polynomial, we get as follows

⇒ 3()³ – 10()² + 9() – 2 = – – – 2

= – – 3 – 2

= – 5

= 0

∴ 3x – 1 is not a factor

⇒ To find the remainder divide second polynomial by first polynomial

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

∴ 3x³ – 10x² + 9x – 2 = (3x – 1)q(x) + c

⇒ 3x³ – 10x² + 9x – 2 –c = (3x – 1)q(x)

⇒ c = ((3x³ – 10x² + 9x – 2) – (3x – 1)) × q(x)

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

⇒

= – – – 2 – 0

∴ c =

is the remainder