What must be subtracted from $x^{4} + 2 x^{3} - 13 x^{2} - 12 x + 21$ so that the result is exactly divisible by $x^{2} - 4 x + 3$ ?

2 x

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3

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3 x

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2 x - 3

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Solution

The correct option is C $2 x - 3$
Given: $f (x) = x^{4} + 2 x^{3} - 13 x^{2} - 12 x + 21$
$g (x) = x^{2} - 4 x + 3$
By division algorithm
$f (x) = q (x) \times g (x) + r (x)$
$g (x) = \frac{f (x) - r (x)}{q (x)}$
So. r(x) must be subtracted so, that result is exactly divisible by $x^{2} - 4 x + 3.$
$2 x - 3$ must be subtracted from f (x), so that result is exactly divisible by $x^{2} - 4 x + 3.$

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