On dividing p(x) by a polynomial x – 1 – x2, the quotient and remainder were (x – 2) and 3 respectively. Find p(x).

Solution

Here,
dividend = p(x)
Divisor, g(x) = (x – 1 – x2)
Quotient, q(x) = (x – 2)
Remainder, r(x) = 3
∵ Dividend = [Divisor × Quotient] + Remainder
∴ P(x) = [g(x) × q(x)] + r(x)
= [(x – 1 – x²) (x – 2)] + 3
= [x² – x – x³ – 2x + 2 + 2x²] + 3
= 3x² – 3x – x³ + 2 + 3
= – x³ + 3x² – 3x + 5