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

4x³ + 7x² - 36x - 63

For x = -3 the value of 4x³ + 7x² - 36x - 63

= 4(-3)³ + 7(-3)² - 36(-3) - 63

= 4(-27) + 7(9) + 108 - 63

= -108 + 63 + 108 - 63

= 0.

Hence, (x + 3) is the factor of 4x³ + 7x² - 36x - 63.

On dividing 4x³ + 7x² - 36x - 63 by (x + 3),

$\begin{array}{l} \phantom{x + 3)}{4x^2 - 5x - 21} \ x + 3\overline{\smash{\big)}4x^3 + 7x^2 - 36x - 63} \ \phantom{x + 3}\underline{\underset{-}{}4x^3 \underset{-}{+} 12x^2} \ \phantom{{x + 3}2x^3+}-5x^2 - 36x \ \phantom{{x + 3}2x^3+}\underline{\underset{+}{-}5x^2 \underset{+}{-} 15x} \ \phantom{{x + 3}{2x^3+}{+2x^2}}-21x - 63 \ \phantom{{x + 3}{2x^3+}{+2x^2}{4}}\underline{\underset{+}{-}21x \underset{+}{-} 63} \ \phantom{{x + 3}{2x^3+}{+2x^2-}{-4x}}\times \end{array}$

we get, quotient = 4x² - 5x - 21

Factorising 4x² - 5x - 21,

= 4x² - 12x + 7x - 21

= 4x(x - 3) + 7(x - 3)

= (4x + 7)(x - 3).

∴ 4x² - 5x - 21 = (4x + 7)(x - 3)

Hence, 4x³ + 7x² - 36x - 63 = (x + 3)(4x + 7)(x - 3).