The polynomial f(x) $= x^{4} + a x^{3} + b x^{2} + c x + d$ has real coefficients and f $(2 i) = f (2 + i) = 0$ . The value of $(a + b + c + d)$ equals to

1

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4

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9

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10

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Solution

The correct option is D $9$
If a polynomial has real coefficients and if some roots are non-real, then the roots occur in pairs of complex conjugates.
The roots are $2 i, - 2 i, 2 + i, 2 - i$ .
Hence, $f (x) = (x + 2 i) (x - 2 i) (x - 2 i) (x - 2 + i)$
$f (1) = (1 + 2 i) (1 - 2 i) (1 - 2 - i) (1 - 2 + i)$
$f (1) = 5 \times 2 = 10$
Also, $f (1) = 1 + a + b + c + d$
$∴ 1 + a + b + c + d = 10$
$\Rightarrow a + b + c + d = 9$

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