Question

If x+2 is a factor of p(x)=ax^3+bx^2+x-6 and p(x)is when divided by x-2: leave a remainder 4. Prove that a=0 and b=2

Answer 1

by remainder theorem  and factor theorem:
when x =-2 : p(-2) = -8a +4b -2-6 = 0 
                         -8a + 4b = 8 .......(1)
when x = 2 : p(2) = 8a + 4b + 2 - 6 = 4
                           8a + 4b  = 8 .......(2)

(1)  + (2)        8b = 16,  therefore b =2
(2) - (1)           16a = 0 therefore a = 0