If $f (x) = x^{4} - 12 x^{3} + 17 x^{2} - 9 x - 7,$ then f(x+3) =

x^{4} - 37 x^{2} - 123 x - 124

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x^{3} - 25 x^{2} - 73 x - 85

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x^{4} - 13 x^{3} - 12 x^{2} - 15

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x^{4} - 47 x^{2} - 15 x - 165

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Solution

The correct option is A $x^{4} - 37 x^{2} - 123 x - 124$
Given:- $f (x) = x^{4} - 12 x^{3} + 17 x^{2} - 9 x - 7$
To find:- $f (x + 3) = ?$
$f (x) = x^{4} - 12 x^{3} + 17 x^{2} - 9 x - 7 (Given)$
Therefore,
$f (x + 3) = {(x + 3)}^{4} - 12 {(x + 3)}^{3} + 17 {(x + 3)}^{2} - 9 (x + 3) - 7$
Using bionomial expansion, we have
$f (x + 3) = (x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81) - 12 (x^{3} + 9 x^{2} + 27 x + 27) + 17 (x^{2} + 6 x + 9) - 9 (x + 3) - 7$
$f (x + 3) = x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81 - 12 x^{3} - 108 x^{2} - 324 x - 324 + 17 x^{2} + 102 x + 153 - 9 x - 27 - 7$
$f (x + 3) = x^{4} + 0 - 37 x^{2} - 123 x - 124$
$f (x + 3) = x^{4} - 37 x^{2} - 123 x - 124$
Hence the correct answer is $x^{4} - 37 x^{2} - 123 x - 124$ .

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