Answer :

Let p(x) = x^{3} + ax^{2} + 5 and q(x) = x^{3} – 2x^{2} + a

As we know by Remainder Theorem,

If a polynomial p(x) is divided by a linear polynomial (x – a) then, the remainder is p(a)

⇒ Remainder of p(x) when divided by x + 2 is p(–2). Similarly, Remainder of q(x) when divided by x + 2 is q(–2)

⇒ p(–2) = (–2)^{3} +a(–2)^{2} + 5

⇒p(–2) = –8 + 4a + 5

⇒p(–2) = –3 + 4a

Similarly, q(–2) = (–2)^{3} – 2(–2)^{2} + a

⇒ q(–2) = –8 –8 + a

⇒ q(–2) = –16 + a

Since they both leave the same remainder, so p(–2) = q(–2)

⇒ –3 + 4a = –16 + a

⇒ –13 = 3a

∴ The value of a is –13/3

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