When the polynomials x^3+2x^2-3ax-1 is divided by x-1, the remainder is A and when the polynomial x^3+ax^2-12x+16 is divided by x+2 the remainder is B. Find the value of a if 2A+B=0.
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using remainder theorem
x = - 1
x^3 + 2x^2 - 5ax - 7 = p
( - 1) ^3 + 2 ( - 1 )^2 - 5 ( - 1 )a - 7 = p
- 1 + 2 - 7 + 5a = p
-6 + 5a = p
when x^2 + ax^2 - 12x + 6 is divided by x - 2
using remainder theorem
x = 2
x^3 + ax^2 - 12x + 6 = q
( 2 )^3 + a ( 2 )^2 - 12 ( 2 ) + 6 = q
8 + 4a - 24 + 6 = q
4a - 10 = q
2p + q = 6
2 ( 5a - 6 ) + (4a - 10 ) = 6
10a - 12 + 4a - 10 = 6
14a - 22 = 6
14a = 28
a = 2
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