$f (x) = x^{4} + 3 x^{2} - 6 x - 2$ then the coefficient of $x^{3}$ in $f (x + 1)$ is

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Solution

The correct option is C 4
Given: $f (x) = x^{4} + 3 x^{2} - 6 x - 2$
To find the coefficient of $x^{3}$ in $f (x + 1)$
Sol: $f (x) = x^{4} + 3 x^{2} - 6 x - 2 f (x + 1) = (x + 1)^{4} + 3 (x + 1)^{2} - 6 (x + 1) - 2 ⟹ f (x + 1) = x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1 + 3 (x^{2} + x + 1) - 6 x - 6 - 2$
As $(a + b)^{4} = a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}$
Hence the coefficient of $x^{3}$ in $f (x + 1)$ is 4

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