Answer :

Let, p (x) = ax3+3x2-3 and q (x) = 2x3-5x+a be the given polynomials.

Now,


R1 = Remainder when p (x) is divided by (x – 4)


= p (4)


= a (4)3 + 3 (4)2 – 3 [Therefore, p (x) = ax3+3x2-3]


= 64a + 48 – 3


R1 = 64a + 45


And,


R2 = Remainder when q (x) is divided by (x – 4)


= q (4)


= 2 (4)3 – 5 (4) + a [Therefore, q (x) = 2x3-5x+a]


= 128 – 20 + a


R2 = 108 + a


(i) Given condition is,


R1 = R2


64a + 45 = 108 + a


63a – 63 = 0


63a = 63


a = 1


(ii) Given condition is R1 + R2 = 0


64a + 45 + 108 + a = 0


65a + 153 = 0


65a = -153


a =


(iii) Given condition is 2R1 – R2 = 0


2 (64a + 45) – (108 + a) = 0


128a + 90 – 108 – a


127a – 18 = 0


127a = 18


a =


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