When divided by x - 3 the polynomials x³ - px² + x + 6 and 2x³ - x² - (p + 3)x - 6 leave the same remainder . Find the value of 'p'.

By remainder theorem, on dividing f(x) by (x - b), remainder = f(b)

∴ On dividing, f(x) = x³ - px² + x + 6 by (x - 3)

Remainder = 3³ - p(3²) + 3 + 6 = 27 - 9p + 9 = 36 - 9p.

∴ On dividing, f(x) = 2x³ - x² - (p + 3)x - 6 by (x - 3)

Remainder = 2(3)³ - 3² - (p + 3)(3) - 6 = 2(27) - 9 - 3p - 9 - 6 = 54 - 9 - 3p - 9 - 6 = 30 - 3p

According to question,

$\Rightarrow 36 - 9p = 30 - 3p \\[0.5em] \Rightarrow 36 - 30 = 9p - 3p \\[0.5em] \Rightarrow 6 = 6p \\[0.5em] \Rightarrow p = 1$

Hence, the value of p is 1.