Using the remainder theorem, factorise each of the following completely :

3x³ + 2x² - 23x - 30

For x = -2 the value of 3x³ + 2x² - 23x - 30,

= 3(-2)³ + 2(-2)² - 23(-2) - 30

= 3(-8) + 2(4) + 46 - 30

= -24 + 8 + 46 - 30

= 54 - 54

= 0.

Hence, (x + 2) is the factor of 3x³ + 2x² - 23x - 30.

On dividing 3x³ + 2x² - 23x - 30 by x + 2,

$\begin{array}{l} \phantom{x + 2)}{3x^2 - 4x - 15} \ x + 2\overline{\smash{\big)}3x^3 + 2x^2 - 23x - 30} \ \phantom{x + 2}\underline{\underset{-}{}3x^3 \underset{-}{+} 6x^2} \ \phantom{{x + 2}2x^3+}-4x^2 - 23x \ \phantom{{x + 2}2x^3+}\underline{\underset{+}{-}4x^2 \underset{+}{-} 8x} \ \phantom{{x + 2}{2x^3+}{+2x^2}}-15x - 30 \ \phantom{{x + 2}{2x^3+}{+2x^2}{4}}\underline{\underset{+}{-}15x \underset{+}{-} 30} \ \phantom{{x + 2}{2x^3+}{+2x^2-}{-4x}}\times \end{array}$

we get, quotient = 3x² - 4x - 15

Factorising, 3x² - 4x - 15

= 3x² - 9x + 5x - 15

= 3x(x- 3) + 5(x - 3)

= (3x + 5)(x - 3).

∴ 3x² - 4x - 15 = (3x + 5)(x - 3).

Hence, 3x³ + 2x² - 23x - 30 = (x + 2)(3x + 5)(x - 3).