Question

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).

Answer 1

Given, p(x) = x³ – 3x² + x + 2 

q(x) = x – 2 and 

r(x) = -2x + 4 

By division algorithm, we know that Dividend = Divisor × Quotient + Remainder 

p(x) = q(x) × g(x) + r(x)

Therefore, x³ – 3x² + x + 2 

= (x – 2) × g(x) + (- 2x + 4)

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

g(x) = \(\frac{x^3-3x^2+3x-2}{x-2}\)

On dividing x³ – 3x² + x + 2, by x – 2, we get

First term of g(x) = x³ / x = x² 

Second term of g(x) = -x² / x = -x

Third term of g(x) = x / x = 1 

Hence, g(x) = x² – x + 1.