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On dividing $x^{3} - 3 x^{2} + x + 2$ by a polynomial $g (x)$ the quotient and remainder were $x - 2$ and - $2 x + 4$ respectively. Find $g (x)$ .

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g (x) = x^{2} - x + 1

g (x) = x^{2} + x + 1

g (x) = x^{2} - x - 1

g (x) = x^{2} + x - 1

answer is A.

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Detailed Solution

Concept- Use the formula

dividend = divisor \times quotient + remainder

to determine

g (x)

. This shall help us solve the given problem.
Euclid division lemma or Euclid division algorithm: This algorithm states that, if we have two positive integers aa and bb, then there exist unique integers

q

and

r

which satisfies the condition

a = bq + r

where

0 \leq r \leq b

.
Now, let us come to the question.
Since,

dividend = divisor \times quotient + remainder

Therefore,

x^{3} - 3 x^{2} + x + 2 = g (x) \times (x - 2) + 4 - 2 x

x^{3} - 3 x^{2} + x + 2 - 4 + 2 x = g (x) \times (x - 2)

x^{3} - 3 x^{2} - 3 x - 2 = g (x) \times (x - 2)

g (x) = \frac{x^{3} - 3 x^{2} - 3 x - 2}{x - 2}

Step 1: So, write

x - 2

three times with some space between all of them.

(x - 2) (x - 2) (x - 2)

Step 2: Get the first term of numerator by multiplying

x^{2}

with

x - 2

x^{2} (x - 2) (x - 2) (x - 2)

Step 3: Balancing the terms with the help of mathematical operations
=

x^{2} (x - 2) - x (x - 2) + 1 (x - 2)

x^{2} (x - 2) - x (x - 2) + 1 (x - 2)

Step 4: Taking

(x - 2)

common.

(x - 2) (x^{2} - x + 1)

Thus,

g (x) = x^{2} - x + 1

.
Hence, the correct option is 1)

g (x) = x^{2} - x + 1

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