Given, 9𝑥² - 4 is a factor of 9𝑥³ - m𝑥² - n𝑥 + 8 :

(a) find the value of m and n using the remainder and factor theorem.

(b) factorise the given polynomial completely.

(a) Simplifying 9𝑥² - 4, we get :

⇒ 9𝑥² - 4

⇒ (3𝑥)² - 2²

⇒ (3𝑥 + 2)(3𝑥 - 2).

We know that,

If (x - a) is a factor of f(x), then f(a) = 0.

Given,

9𝑥² - 4 is a factor of 9𝑥³ - m𝑥² - n𝑥 + 8.

∴ (3𝑥 + 2) and (3𝑥 - 2) are the factors of 9𝑥³ - m𝑥² - n𝑥 + 8.

⇒ 3𝑥 + 2 = 0

⇒ 3𝑥 = -2

⇒ 𝑥 = $-\dfrac{2}{3}$

Substituting x = $-\dfrac{2}{3}$ in 9𝑥³ - mx² - nx + 8, we get remainder = 0.

$\Rightarrow 9 \times \Big(-\dfrac{2}{3}\Big)^3 - m \times \Big(-\dfrac{2}{3}\Big)^2 - n \times \Big(-\dfrac{2}{3}\Big) + 8 = 0 \\[1em] \Rightarrow 9 \times -\dfrac{8}{27} - m \times \dfrac{4}{9} + \dfrac{2n}{3} + 8 = 0 \\[1em] \Rightarrow -\dfrac{8}{3} - \dfrac{4m}{9} + \dfrac{2n}{3} + 8 = 0 \\[1em] \Rightarrow \dfrac{-24 - 4m + 6n + 72}{9} = 0 \\[1em] \Rightarrow \dfrac{6n - 4m + 48}{9} = 0 \\[1em] \Rightarrow 6n - 4m + 48 = 0 \\[1em] \Rightarrow 2(3n - 2m + 24) = 0 \\[1em] \Rightarrow 3n - 2m + 24 = 0 \\[1em] \Rightarrow 3n - 2m = -24 \text{ ……….(1)}$

⇒ 3𝑥 - 2 = 0

⇒ 3𝑥 = 2

⇒ 𝑥 = $\dfrac{2}{3}$

Substituting 𝑥 = $\dfrac{2}{3}$ in 9𝑥³ - m𝑥² - n𝑥 + 8, we get remainder = 0.

$\Rightarrow 9 \times \Big(\dfrac{2}{3}\Big)^3 - m \times \Big(\dfrac{2}{3}\Big)^2 - n \times \Big(\dfrac{2}{3}\Big) + 8 = 0 \\[1em] \Rightarrow 9 \times \dfrac{8}{27} - m \times \dfrac{4}{9} - \dfrac{2n}{3} + 8 = 0 \\[1em] \Rightarrow \dfrac{8}{3} - \dfrac{4m}{9} - \dfrac{2n}{3} + 8 = 0 \\[1em] \Rightarrow \dfrac{24 - 4m - 6n + 72}{9} = 0 \\[1em] \Rightarrow \dfrac{96 - 6n - 4m}{9} = 0 \\[1em] \Rightarrow 96 - 6n - 4m = 0 \\[1em] \Rightarrow 4m + 6n = 96 \\[1em] \Rightarrow 2(2m + 3n) = 96 \\[1em] \Rightarrow 2m + 3n = \dfrac{96}{2} \\[1em] \Rightarrow 2m + 3n = 48 \text{ ……….(2)}$

Adding equation (1) and (2), we get :

⇒ 3n - 2m + 2m + 3n = -24 + 48

⇒ 6n = 24

⇒ n = $\dfrac{24}{6}$

⇒ n = 4.

Substituting value of n in equation (1), we get :

⇒ 3(4) - 2m = -24

⇒ 12 - 2m = -24

⇒ -2m = -24 - 12

⇒ -2m = -36

⇒ m = $\dfrac{-36}{-2}$ = 18.

Hence, m = 18 and n = 4.

(b) Substituting value of m and n in 9𝑥³ - m𝑥² - n𝑥 + 8, we get :

9𝑥³ - 18𝑥² - 4𝑥 + 8

Dividing 9𝑥³ - 18𝑥² - 4𝑥 + 8 by 9𝑥² - 4, we get :

$\begin{array}{l} \phantom{9x^2 - 4)}{\quad x - 2} \ 9x^2 - 4\overline{\smash{\big)}\quad 9x^3 - 18x^2 - 4x + 8} \ \phantom{9x^2 - 4)}\phantom{)}\underline{\underset{-}{+}9x^3 \phantom{- 18x^2} \underset{+}{-}4x} \ \phantom{{9x^2 - 4}{+9x^3 - }}-18x^2 \phantom{- 4x} + 8 \ \phantom{{9x^2 - 4)}{+9x^3 - }}\underline{\underset{+}{-}18x^2 \phantom{- 4x} \underset{-}{+} 8} \ \phantom{{9x^2 - 4)}{x^3-2x^{2}(31)}{x}}\times \end{array}$

∴ 9𝑥³ - 18𝑥² - 4𝑥 + 8 = (9𝑥² - 4)(𝑥 - 2)

= (3𝑥 + 2)(3𝑥 - 2)(𝑥 - 2).

Hence, factors are (3𝑥 + 2), (3𝑥 - 2) and (𝑥 - 2).