In each of the following two polynomials, find the value of a, if x-a is a factor:
(i) x6-ax5+x4-ax3+3x-a+2.
(ii) x5-a2x3+2x+a+1.
(i) Let f (x) = x6-ax5+x4-ax3+3x-a+2 be the given polynomial
From factor theorem,
If (x – a) is a factor of f (x) then f (a) = 0 [Therefore, x – a = 0, x = a]
f (a) = 0
(a)6 – a (a)5 + (a)4 – a (a)3 + 3 (a) – a + 2 = 0
a6 – a6 + a4 – a4 + 3a – a + 2 = 0
2a + 2 = 0
a = -1
Hence, (x – a) is a factor f (x) when a = -1.
(ii) Let, f (x) = x5-a2x3+2x+a+1 be the given polynomial
From factor theorem,
If (x – a) is a factor of f (x) then f (a) = 0 [Therefore, x – a = 0, x = a]
f (a) = 0
(a)5 – a2 (a)3 + 2 (a) + a + 1 = 0
a5 – a5 + 2a + a + 1 = 0
3a + 1 = 0
3a = -1
a =
Hence, (x – a) is a factor f (x) when a = .