If x = -2 is the root of the equation root 2 (x + p) = 0 and is also the zero of the polynomial px to the power 2 + kx + 2 root 2 then find the value of k.

Given, x  = -2 is a root of the equation √2(x + p) = 0.

 ⇒ √2(-2 + p) = 0

⇒ p = 2

Again, x = -2 is also a zero of the polynomial say, f(x) = px² + kx + 2√2

⇒ f(-2) = 0

⇒ p(-2)² + k(-2) + 2√2 = 0

⇒ 4(2) -2k + 2√2 = 0

⇒ 8 -2k + 2√2 = 0

⇒ 8 + 2√2 = 2k

⇒ k = 4 + √2