What will the quotient and remainder be on division of ax² + bx + c by px³ + qx² + rx + s, p≠0?

quotient = 0 and remainder = ax² + bx + c

Let p(x) = ax² + bx + c and g(x) = px³ + qx² + rx + s , p≠0

Clearly the degree of divisor is greater than the degree of the dividend

i.e. deg (p(x)) < deg(g(x))

∴ deg (p(x)) < deg (g(x)) × deg (q(x)) where deg (q(x)) can take any value …(1)

By division algorithm,

p(x) = g(x).q(x) + r(x)

Hence the degree in the L.H.S is equal to the degree of R.H.S

we know that the degree of r(x) will always be less than or equal to p(x)

⇒ deg (p(x)) = deg(g(x))× deg(q(x)) …(2)

(1) and (2) do not go hand in hand unless q(x) = 0

∴ p(x) = g(x).q(x) + r(x)

= p(x) = 0 + r(x)

⇒ p(x) = r(x)

∴ The quotient = 0 and the remainder is same as the dividend.