Find the value of k, if x - 1 is a factor of p(x) in each of the following cases:

(i) p(x) = x² + x + k

(ii) p(x) = 2x² + kx + $\sqrt{2}$

(iii) p(x) = kx² - $\sqrt{2}$x + 1

(iv) p(x) = kx² - 3x + k

(i) p(x) = x² + x + k

⇒ x - 1 = 0

⇒ x = 1

Putting x = 1 we get,

p(1) = (1)² + 1 + k

= 1 + 1 + k

⇒ 2 + k = 0

⇒ k = -2

So the value of k = -2

(ii) p(x) = 2x² + kx + $\sqrt{2}$

⇒ x - 1 = 0

⇒ x = 1

Putting x = 1 we get,

p(1) = 2(1)² + k(1) + $\sqrt{2}$

⇒ 2 + k + $\sqrt{2}$

⇒ k = -2 -$\sqrt{2}$

⇒ k = -(2 + $\sqrt{2}$)

So the value of k = -(2 + $\sqrt{2}$)

(iii) p(x) = kx² - $\sqrt{2}$x + 1

⇒ x - 1 = 0

⇒ x = 1

Putting x = 1 we get,

p(1) = k(1)² - $\sqrt{2}$(1) + 1

⇒ k - $\sqrt{2}$ + 1 = 0

⇒ k = $\sqrt{2}$ - 1

So the value of k = $\sqrt{2}$ - 1

(iv) p(x) = kx² - 3x + k

⇒ x - 1 = 0

⇒ x = 1

Putting x = 1 we get,

p(1) = k(1)² -3(1) + k

= k - 3 + k

⇒ 2k - 3 = 0

⇒ 2k = 3

⇒ k = $\dfrac{3}{2}$

So the value of k = $\dfrac{3}{2}$