Apply 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) = 10x4 + 17x3 – 62x2 + 30x – 3
g(x) = 2x2 + 7x + 1

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q(x) = 5x2 – 9x – 2, r(x) = –1

q(x) = 5x2 + 9x + 2, r(x) = –1

q(x) = 5x2 + 9x – 2, r(x) = 1

q(x) = 5x2 - 9x + 2, r(x) = –1

answer is A.

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Detailed Solution

f(x) = 10x4 + 17x3 – 62x2 + 30x – 3
g(x) = 2x2 + 7x + 1
Let q(x) = ax2 + bx + c and r(x) = l
By division algorithm formula →
⇒ f(x) = g(x) × q + x
⇒10x4 + 17x3 – 62x2 + 30x – 3 = ax2 + bx + c × 2x2 + 7x +1 + k
⇒10x4 + 17x3 – 62x2 + 30x – 3 = 4ax4 + 7ax3 + ax2 + 2bx3 + 7bx2 + bx + 2cx2 + 7cx + c + k
⇒10x4 + 17x3 – 62x2 + 30x – 3 = 4ax4 + (7a + 2b)x3 + (a + 7b + 2c)x2 + (b + 7c)x + c + x
By comparing coefficient of x4 →
⇒ 2a = 10 ⇒ a = 5
By comparing coefficient of x3 →
⇒ 17 = 7a + 2b
⇒ 2b + 7 × 5 = 17 ⇒ b = –9
By comparing coefficient of x2 →
⇒ –62 = a + 7b + 2a
⇒ –62 = 5 + (7 × – 9) + 2a
⇒ c== 2 By comparing constant → c + k = –3
⇒ –2 + k = 3 ⇒ k = –1
Hence quotient q(x) = ax2 + bx + c = 5x2 – 9x – 2
Remainder r(x) = –1

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