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Algebra of Derivatives

The value of ...

Question

The value of $b$ so that $x^{4} - 3 x^{3} + 5 x^{2} - 33 x + b$ is divisible by $x^{2} - 5 x + 6$ , is:

45

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48

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51

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54

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Solution

The correct option is D $54$
Since the given polynomial is exactly divisible by $x^{2} - 5 x + 6 = (x - 2) (x - 3)$ , it should be exactly divisible by $x - 2$ and $x - 3$ as well.
$∴$ By remainder theorem, at $x = 2; 16 - 24 + 20 - 66 + b = 0 \Rightarrow b = 54$

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