If (x + 3) and (x - 4) are factors of x³ + ax² - bx + 24, find the values of a and b. With these values of a and b, factorise the given expression.

By factor theorem (x - b) is a factor of f(x), if f(b) = 0.

f(x) = x³ + ax² - bx + 24

Given, (x + 3) or (x - (-3) and (x - 4) are factors of f(x)

∴ f(-3) = 0 and f(4) = 0.

For, f(-3) = 0

$\Rightarrow (-3)^3 + a(-3)^2 - b(-3) + 24 = 0 \\[0.5em] \Rightarrow -27 + 9a + 3b + 24 = 0 \\[0.5em] \Rightarrow 9a + 3b - 3 = 0 \\[0.5em] \Rightarrow 9a + 3b = 3$

On dividing equation by 3,

$\Rightarrow 3a + b = 1 \\[0.5em] b = 1 - 3a \text{ \space (Equation 1) }$

For f(4) = 0

$\Rightarrow (4)^3 + a(4)^2 - b(4) + 24 = 0 \\[0.5em] \Rightarrow 64 + 16a - 4b + 24 = 0 \\[0.5em] \Rightarrow 16a - 4b + 88 = 0 \\[0.5em] \Rightarrow 16a - 4b = -88$

On dividing equation by 4,

$\Rightarrow 4a - b = -22$

Putting value of b = 1 - 3a from equation 1,

$\Rightarrow 4a - 1 + 3a = -22 \\[0.5em] \Rightarrow 7a - 1 = -22 \\[0.5em] \Rightarrow 7a = -22 + 1 \\[0.5em] \Rightarrow 7a = -21 \\[1em] \Rightarrow a = -\dfrac{21}{7} \\[1em] \Rightarrow a = -3 \\[0.5em] \text{ and } b = 1 - 3a = 1 - 3(-3) = 1 + 9 = 10$

Now putting a = -3 and b = 10 in f(x),

f(x) = x³ - 3x² - 10x + 24

Since, (x + 3) and (x - 4) is a factor of f(x), hence (x + 3)(x + 4) is also the factor

$(x + 3)(x - 4) = x^2 + 3x - 4x - 12 \\[0.5em] = x^2 - x - 12$

On dividing, f(x) by x² - x - 12,

$\begin{array}{l} \phantom{x^2 - x - 12)}{x - 2} \ x^2 - x - 12\overline{\smash{\big)}x^3 - 3x^2 - 10x + 24} \ \phantom{x^2 - x - 12}\underline{\underset{-}{ }x^3 \underset{+}{-} x^2 \underset{+}{-} 12x} \ \phantom{{x^2 - x - 12}{x^3+}}-2x^2 + 2x + 24 \ \phantom{{x^2 - x - 12}x^3+}\underline{\underset{+}{-}2x^2 \underset{-}{+} 2x \underset{-}{+} 24} \ \phantom{{x^2 - x - 12}{2x^3+}{+2x^2-}}\times \end{array}$

we get (x - 2) as quotient and remainder = 0.

$\therefore x^3 - 3x^2 - 10x + 24 = (x - 2)(x^2 - x - 12). \\[0.5em] = (x - 2)(x^2 - 4x + 3x - 12) \\[0.5em] = (x - 2)(x(x - 4) + 3(x - 4)) \\[0.5em] = (x - 2)(x + 3)(x - 4)$

Hence, value of a = -3 and b = 10; x³ - 3x² - 10x + 24 = (x - 2)(x + 3)(x - 4).