What should be subtracted from $x^{3} - 7 x^{2} + 17 x + 17$ so that the difference is a multiple of $x - 3$ ?

5

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32

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7

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43

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Solution

The correct option is B $32$
Let $P (x) = x^{3} - 7 x^{2} + 17 x + 17$
Let us subtract $a$ from $P (x)$ to make it a multiple by $x - 3$
$P (x) - a = x^{3} - 7 x^{2} + 17 x + 17 - a$
This is divisible by $(x - 3)$
Hence, when $x = 3$ , $P (x) - a = 0$
$\Rightarrow 3^{3} - 7 \times 3^{2} + 17 \times 3 + 17 - a = 0.$
$∴$ $27 - 63 + 51 + 17 - a = 0.$
$∴$ $78 + 17 - 63 - a = 0.$
$∴$ $a = 32.$
Hence, we need to subtract 32 from $x^{3} - 7 x^{2} + 17 x + 17$ so that the difference is divisible by $x - 3$

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