The polynomials x³ + x² – 4x + a and 2x³ + ax² + 3x – 3 when divided by x - 2 give the same remainder. Then find the value of a.

Let p(x) = x³ + x² – 4x + a and q(x) = 2x³ + ax² + 3x – 3

We have to find remainders when p(x) is divided by x – 2 and when q(x) is divided by x – 2 and equate them as it is given that both are equal

Now we will find remainder using remainder theorem which states that

If a polynomial t(x) is divided by a polynomial x – b then the remainder is t(b)

Compare x – 2 with x – b we have b = 2

Let us first find remainder when p(x) is divided by x – 2

Substitute 2 in p(x)

⇒ p(2) = 2³ + 2² – 4(2) + a

⇒ p(2) = 8 + 4 – 8 + a

⇒ p(2) = 4 + a …(i)

Now let us find remainder when q(x) is divided by x – 2

Substitute 2 in q(x)

⇒ q(2) = 2(2)³ + a(2)² + 3(2) – 3

⇒ q(2) = 2(8) + 4a + 6 – 3

⇒ q(2) = 16 + 4a + 3

⇒ q(2) = 19 + 4a …(ii)

Equate (i) and (ii)

⇒ 4 + a = 19 + 4a

⇒ 4a – a = 4 - 19

⇒ 3a = - 15

⇒ a = - 5

Hence value of a is - 5