On dividing the polynomial f(x) = x³ – 3x² + x + 2 by the polynomial g(x), quotient q(x) and remainder f(x) are respectively obtained as x – 2 and –2x + 4. Find the polynomial g(x).

Given:

Dividend = f(x) = x³ – 3x² + x + 2

Divisor = g(x)

Quotient = x – 2

Remainder = -2x + 4

There is an important relation between dividend, divisor, quotient and remainder which is as follows:

Dividend = Divisor × Quotient + Remainder

x³ – 3x² + x + 2 = g(x) × (x-2) + (4-2x)

g(x) × (x-2) = x³ – 3x² + x + 2 – (4 – 2x)

= x³ – 3x² + x + 2 – 4 + 2x

g(x) × (x-2) = x³ – 3x² + 3x – 2

Therefore g(x) = (x³ – 3x² + 3x – 2) / (x-2)

Therefore the g(x) = x² – x – 1