Q. 1 B3.7( 12 Votes )
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.
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