When x³ - 3x² + x + 2 is divided by polynomial g(x), then quotient and remainder are x - 2 and - 2x + 4, respectively, g(x) is:

Question: When x³ - 3x² + x + 2 is divided by polynomial g(x), then quotient and remainder are x - 2 and - 2x + 4, respectively, g(x) is:

Solution:
According to the division algorithm,
Dividend = Divisor × Quotient + Remainder

We have,
Dividend = x³ - 3x² + x + 2,
Divisor = g(x),
Quotient = x - 2
and Remainder = -2x + 4

Put the given values in the below equation and simplify it, to get the value of g (x).

Dividend = Divisor × Quotient + Remainder
(x³ - 3x² + x + 2) = g(x) × (x - 2) + (- 2x + 4)
⇒ (x³ - 3x² + x + 2) - (- 2x + 4) = g(x) × (x - 2)
⇒ (x³ - 3x² + x + 2x + 2 - 4) = g(x) × (x - 2)
⇒ (x³ - 3x² + 3x - 2) = g(x) × (x - 2)
⇒ g(x) = (x³ - 3x² + 3x - 2)/(x - 2)

Therefore, g(x) = x² - x + 1