On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x–2 and –2x+4, respectively. Find g(x).

On dividing x³ – 3x² + x + 2 by a polynomial g(x), the quotient and remainder were x–2 and –2x+4, respectively. Find g(x).

according to division algorithm

Dividend = Divisor × Quotient+ Remainder

p(x)         = g(x)      × q(x)          + r(x),

plug the value in formula we get

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

Add 2x and subtract 4 both side we get

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

simplify and divide by x/2 we get

(x² – 3x² + 3x– 2)/(x-2) = g(x)

 So we get g(x)  = x² – x +1