If x-2 is a factor of each of the following two polynomials, find the values of a in each case.
(i) x3-2ax2+ax-1
(ii) x5-3x4-ax3+3ax2+2ax+4
(i) Let, f (x) = x3-2ax2+ax-1 be the given polynomial
From factor theorem,
If (x – 2) is a factor of f (x) then f (2) = 0 [Therefore, x – 2 = 0, x = 2]
f (2) = 0
(2)3 – 2 a (2)2 + a (2) – 1 = 0
8 – 8a + 2a – 1 = 0
7 – 6a = 0
6a = 7
a = 
Hence, (x – 2) is a factor of f (x) when a = 
.
(ii) Let f (x) = x5-3x4-ax3+3ax2+2ax+4 be the given polynomial
From factor theorem,
If (x – 2) is a factor of f (x) then f (2) = 0 [Therefore, x – 2= 0, x = 2]
f (2) = 0
(2)5 – 3 (2)4 – a (2)3 + 3 a (2)2 + 2 a (2) + 4 = 0
32 – 48 – 8a + 12a + 4a + 4 = 0
-12 + 8a = 0
8a = 12
a = 
Hence, (x – 2) is a factor of f (x) when a = 
.
23