Use the Remainder theorem to find which of the following is a factor of 2x³ + 3x² - 5x - 6.

(i) x + 1

(ii) 2x - 1

(iii) x + 2

(i) x + 1 = 0 ⇒ x = -1

Required remainder = Value of given polynomial 2x³ + 3x² - 5x - 6 at x = -1.

∴ Remainder = 2(-1)³ + 3(-1)² - 5(-1) - 6

= 2(-1) + 3(1) + 5 - 6

= -2 + 3 + 5 - 6

= 8 - 8

Since, remainder = 0

∴ x + 1 is a factor of 2x³ + 3x² - 5x - 6

(ii) 2x - 1 = 0 ⇒ x = $\dfrac{1}{2}$

Required remainder = Value of given polynomial 2x³ + 3x² - 5x - 6 at x = $\dfrac{1}{2}$.

$\therefore \text{ Remainder} = 2\Big(\dfrac{1}{2}\Big)^3 + 3\Big(\dfrac{1}{2}\Big)^2 - 5\Big(\dfrac{1}{2}\Big) - 6 \\[1em] = 2 \times \dfrac{1}{8} + 3 \times \dfrac{1}{4} - \dfrac{5}{2} - 6 \\[1em] = \dfrac{1}{4} + \dfrac{3}{4} - \dfrac{5}{2} - 6 \\[1em] = \dfrac{1 + 3 - 10 - 24}{4} \\[1em] = -\dfrac{30}{4} \\[1em] = -\dfrac{15}{2}.$

Since, remainder ≠ 0

∴ 2x - 1 is not a factor of 2x³ + 3x² - 5x - 6.

(iii) x + 2 = 0 ⇒ x = -2

Required remainder = Value of given polynomial 2x³ + 3x² - 5x - 6 at x = -2.

∴ Remainder = 2(-2)³ + 3(-2)² - 5(-2) - 6

= 2(-8) + 3(4) + 10 - 6

= -16 + 12 + 10 - 6

= 22 - 22

= 0.

Since, remainder = 0

∴ x + 2 is a factor of 2x³ + 3x² - 5x - 6.