Apply a division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in each of the following.
f(x) = 15x3 – 20x2 + 13x – 12
g(x) = x2 – 2c + 2

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q(x) = 15x + 10, r(x) = 32

q(x) = 15x + 10, r(x) =–32

q(x) = 15x – 10, r(x) = –32

q(x) = 15x – 10, r(x) = 32

answer is B.

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Detailed Solution

f(x) = 15x3 – 20x2 + 13x – 12
g(x) = x2 – 2c + 2
Let q(x) = ax + b
R(x) = k
By division algorithm formula →
⇒ f(x) = g(x)×q +r
⇒15x3 – 20x2 + 13x – 12 = (ax + bx2 –2x +2) + k
⇒ 15x3 – 20x2 + 13x – 12 = ax3 – 2ax2 + 2ax + bx2 – 2bx + b + k
⇒ 15x3 – 20x2 + 13x – 12 = ax3 + (b – 2a)x2 +(2a – 2b)x + 2b + k
By considering coefficient of x3 →
⇒ a = 15
By considering Coefficient of x2 →
⇒ b – 2a = –20
⇒ b – 2 × 15 = –20
⇒ b = 10
By comparing constant → 2b + k = –12
⇒ 2 × 10 + k = –12
⇒ k = –32
Hence, quotient q(x) = ax + b = 15x + 10
Remainder r(x) = k = –32

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