If (x - 2) is a factor of 2x³ - x² - px - 2,

(i) find the value of p

(ii) with the value of p, factorize the above expression completely.

(i) Let f(x) = 2x³ - x² - px - 2

Since, (x − 2) is the factor, f(2) = 0.

⇒ 2(2)³ - (2)² - p(2) - 2 = 0

⇒ 16 - 4 - 2p - 2 = 0

⇒ 10 - 2p = 0

⇒ 2p = 10

⇒ p = $\dfrac{10}{2}$

⇒ p = 5.

Hence, the value of p = 5.

(ii) f(x) = 2x³ - x² - 5x - 2

Now, dividing f(x) by (x - 2), we get :

$\begin{array}{l} \phantom{x -1 ]3)}{2x^2 + 3x + 1} \ x - 2\overline{\smash{\big)}2x^3 - x^2 - 5x - 2} \ \phantom{x - 2}\underline{\underset{-}{}2x^3 \underset{+}{-}4x^2} \ \phantom{{x - 2}x^,.3-}3x^2 - 5x \ \phantom{{x -l2}fx^l.\space}\underline{\underset{-}{+}3x^2 \underset{+}{-}6x} \ \phantom{{x - 2]euo[ki]}x^3okk\space}{x - 2} \ \phantom{{x - 2}x^3o;llk]lmk\space}\underline{\underset{-}{+}x\underset{+}{-}2} \ \phantom{{x - 2}{x^,jo-k2\space}{-9x}}\times \end{array}$

2x³ - x² - 5x - 2 = (x - 2)(2x² + 3x + 1)

= (x - 2)(2x² + 2x + x + 1)

= (x - 2)[2x(x + 1) + 1(x + 1)]

= (x - 2)(2x + 1)(x + 1)

Hence, 2x³ - x² - 5x - 2 = (x - 2)(2x + 1)(x + 1).