It is given that (x − 2) is a factor of polynomial 2x³ − 7x² + kx − 2.

Find:

(a) the value of ‘k’.

(b) hence, factorise the resulting polynomial completely.

(a) Since (x − 2) is a factor of 2x³ − 7x² + kx − 2.

Thus, on substituting x = 2, in 2x³ − 7x² + kx − 2, the remainder will be equal to zero.

⇒ 2(2)³ − 7(2)² + k(2) − 2 = 0

⇒ 16 − 28 + 2k − 2 = 0

⇒ −14 + 2k = 0

⇒ 2k = 14

⇒ k = $\dfrac{14}{2}$

⇒ k = 7.

Hence, k = 7.

(b) Substituting k = 7 in 2x³ − 7x² + kx − 2, we get :

Polynomial : 2x³ − 7x² + 7x − 2.

Dividing 2x³ − 7x² + 7x − 2 by x - 2, we get :

$\begin{array}{l} \phantom{x - 31}{\quad2x^2 - 3x + 1} \ x - 2\overline{\smash{\big)}\quad 2x^3 − 7x^2 + 7x − 2} \ \phantom{x^2 + 4}\phantom(\underline{\underset{-}{+}2x^3 \underset{+}{-}4x^2} \ \phantom{x^2 + 3x - 7)} - 3x^2 + 7x \ \phantom{x^2 + 3x - 5))}\underline{\underset{+}{-}3x^2 \underset{-}{+}6x} \ \phantom{x^2 + 3x - 5) + 24x ())}x - 2 \ \phantom{{x^2 + 3x - 5) + 24 +)}}\underline{\underset{-}{+}x \underset{+}{-}2} \ \phantom{{x^2 + 3x - 54)} + 2x + 7} \times \end{array}$

⇒ 2x³ − 7x² + 7x − 2 = (x - 2)(2x² - 3x + 1)

⇒ 2x³ − 7x² + 7x − 2 = (x - 2)[2x² - 2x - x + 1]

⇒ 2x³ − 7x² + 7x − 2 = (x - 2)[2x(x - 1) - 1(x - 1)]

⇒ 2x³ − 7x² + 7x − 2 = (x - 2)(2x - 1)(x - 1).

Hence, 2x³ − 7x² + 7x − 2 = (x - 2)(2x - 1)(x - 1).