Answer :

Given, a pair of polynomial as 2x – 1, 3x^{3} – x^{2} – 8x + 6

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

⇒ To check 2x – 1 is a factor of 3x^{3} – x^{2} – 8x + 6 we must substitute x = in the second polynomial, we get as follows

⇒ 3()^{3} – ()^{2} – 8() + 6 = + 6 = – + 6 = + 2 = 0

∴ 2x – 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

∴ 3x^{3} – x^{2} – 8x + 6 = (2x – 1)q(x) + c

⇒ 3x^{3} – x^{2} – 8x + 6 –c = (2x – 1)q(x)

⇒ c = ((3x^{3} – x^{2} – 8x + 6) – (2x – 1)) × q(x)

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

⇒ c = (3()^{3} – ()^{2} – 8() + 6 – 2() + 1) × q(1)

= – + 6

∴ c =

is the remainder

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