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.
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(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 = $\frac{14}{2}$
⇒ 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).