When 3x² - 5x + p is divided by (x - 2), the remainder is 3. Find the value of p. Also factorise the polynomial 3x² - 5x + p - 3.

By remainder theorem, on dividing f(x) by (x - a), the remainder left is f(a).

f(x) = 3x² - 5x + p

∴ On dividing f(x) by (x - 2), Remainder = f(2)

Given, Remainder = 3

∴ f(2) = 3

$\Rightarrow 3(2)^2 - 5(2) + p = 3 \\[0.5em] \Rightarrow 12 - 10 + p = 3 \\[0.5em] \Rightarrow p + 2 = 3 \\[0.5em] \Rightarrow p = 3 - 2 \\[0.5em] p = 1$

Putting value of p = 1 in 3x² - 5x + p - 3,

$3x^2 - 5x + 1 - 3 \\[0.5em] = 3x^2 - 5x - 2 \\[0.5em] = 3x^2 - 6x + x - 2 \\[0.5em] = 3x(x - 2) + 1(x - 2) \\[0.5em] (3x + 1)(x - 2)$

Hence, the value of p is 1 and the factors are (3x + 1) and (x - 2).