Pamela factorized the following polynomial :

2x³ + 3x² - 3x - 2

She found the result as (x + 2)(x - 1)(x - 2). Using remainder and factor theorem, verify whether her result is correct. If incorrect, give the correct result.

Dividing the polynomial f(x) = 2x³ + 3x² - 3x - 2 by (x + 2), we get :

⇒ x + 2 = 0

⇒ x = -2

f(-2) = 2(-2)³ + 3(-2)² - 3(-2) - 2

= 2(-8) + 3(4) + 6 - 2

= -16 + 12 + 6 - 2

= -18 + 18

= 0.

Dividing the polynomial f(x) = 2x³ + 3x² - 3x - 2 by (x - 1), we get :

⇒ x - 1 = 0

⇒ x = 1

f(1) = 2(1)³ + 3(1)² - 3(1) - 2

= 2(1) + 3(1) - 3 - 2

= 2 + 3 - 3 - 2

= 0.

Dividing the polynomial f(x) = 2x³ + 3x² - 3x - 2 by (x - 2), we get :

⇒ x - 2 = 0

⇒ x = 2

f(2) = 2(2)³ + 3(2)² - 3(2) - 2

= 2(8) + 3(4) - 6 - 2

= 16 + 12 - 6 - 2

= 20.

Since, f(2) ≠ 0

∴ x - 2 does not divide the polynomial 2x³ + 3x² - 3x - 2.

Dividing the polynomial f(x) by (x + 2)(x - 1) or by (x² + x - 2), we get :

$\begin{array}{l} \phantom{x^2 + x - 2)}{\quad 2x + 1} \ x^2 + x - 2\overline{\smash{\big)}\quad 2x^3 + 3x^2 - 3x - 2} \ \phantom{x^2 + x - 2)}\phantom{)}\underline{\underset{-}{+}2x^3 \underset{-}{+}2x^2 \underset{+}{-}4x} \ \phantom{{x^2 + x - 2}2x^3 + 20x^3}x^2 + x - 2 \ \phantom{{x^2 + x - 2)}2x^3 + x^3}\underline{\underset{-}{+}x^2 \underset{-}{+} x \underset{+}{-} 2} \ \phantom{{x^2 + x - 2)}{x^3-2x^{2}(31)}{x}}\times \end{array}$

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

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

Hence, 2x³ + 3x² - 3x - 2 = (x + 2)(x - 1)(2x + 1).