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Question

On dividing $$f(x)=3x^3+x^2+2x+5$$ by a polynomial $$g(x)=x^2+2x+1$$, the remainder $$r(x)=9x+10$$. Find the quotient polynomial $$q(x)$$


A
q(x)=x15
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B
q(x)=3x5
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C
q(x)=4x18
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D
None of these
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Solution

The correct option is B $$q(x)=3x-5$$
By remainder theorem,
$$f(x)=q(x)g(x)+r(x)$$
$$\therefore 3x^3+x^2+2x+5=q(x)(x^2+2x+1)+9x+10$$
$$\Rightarrow q(x)(x^2+2x+1)=3x^3+x^2-7x-5$$
Now,
$$x^2+2x+1)\overline {3x^3+x^2-7x-5}$$ ( $$3x-5=q(x)$$
                       $$\underline {\underset {-}{3}x^3\underset {-}{+}6x^2\underset {-}{+}3x}$$
                       $$-5x^2-10x-5$$
                       $$\underline {\underset {+}{-}{5}x^2\underset {+}{-}10x\underset {+}{-}5}$$
                              $$0$$
Hence, $$q(x)=3x-5$$

Mathematics

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