Q6 of 86 Page 42

If the polynomials x3 + ax2 + 5 and x3 – 2x2 + a are divided by (x + 2) leave the same remainder, find the value of a.

Let p(x) = x3 + ax2 + 5 and q(x) = x3 – 2x2 + 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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