Let p(x) = xn + an
The zero of x + a is - a. | x + a = 0 ⇒ x = - a
Now,
p(- a) = (- a)n + an = (- 1)nan + an
     = (- 1)an + anÂ
  Â
 n is an odd positive interger
  (-1)n = - 1
= - an + an = 0
∴ By Factor Theorem, x + a is a factor of xn + an for any odd positive integer n.
Let p(x) = 5x3 - x2 + 4x + b
1 - 5x = 0
 The zero of 1 - 5x isÂ
If p(x) is divisivle by 1 - 5x, thenÂ
                     | By Factor Theorem
Let p (x) = x4Â + ax3Â - 3x2Â + 2x + b
If (x + 1) and (x - 1) are factors of p(x), then by factor theorem,
p(-1) = 0 ...(1) | x + 1 = 0 ⇒ x = -1
and    p(1) = 0 ...(2) | x - 1 = 0 ⇒ x = 1
Now,
P(-1) = 0
⇒ (-1)4 + a(-1)3 - 3(-1)2 + 2 (-1) + b = 0
⇒ 1 - a - 3 - 2 + b = 0
⇒    -a + b = 4    ...(3)
and    p(1) = 0
⇒ (1)4 + a(1)3 - 3(1)2 + 2 (1) + b = 0
⇒ 1 + a - 3 + 2 + b = 0
⇒    a + b = 0    ...(4)
Solving (3) and (4), we get
a = -2,b = 2
Let p(x) = 2x3Â + ax2Â + 11x + a + 3
If p(x) is exactly divisible by 2x - 1, then by factor theorem,
Hence, the required polynomial is
2x3Â - 7x2Â + 11x - 7 + 3
or 2x3Â - 7x2Â + 11x - 4
x2Â + 2x - 3
= x2Â + 3x - x - 3
= x(x + 3) - 1 (x + 3)
= (x + 3) (x - 1)
Let p(x) = x4Â + 2x3Â - 2x2Â + 2x - 3
We see that
p(-3) = (-3)4Â + 2(-3)3Â - 2(-3)2Â + 2(-3) - 3
= 81 - 54 - 18 - 6 - 3
= 0
Hence by converse of factor theorem, (x + 3) is a factor of p(x).
Also, we see that
p(1) = (1)4Â + 2(1)3Â - 2(1)2Â + 2(1) - 3
= 0
Hence by converse of factor theorem, (x - 1) is a factor of p(x).
From above, we see that
(x + 3) (x - 1), i.e., x2Â + 2x - 3 is a factor of p(x)
⇒ p(x) is exactly divisible by (x2 + 2x - 3).