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Exponents with like Bases

If n>1, the...

Question

If $n > 1$ , then ${(1 + x)}^{n} - n x - 1$ is divisible by

x

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x^{2}

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x^{3}

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x^{4}

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Solution

The correct options are
A $x$
B $x^{2}$
$(1 + x)^{n} - n x - 1$
$= [1 +^{n} C_{1} x +^{n} C_{2} x^{2} + . . .^{n} C_{n} x^{n}] - n x - 1$
$= [1 + n x +^{n} C_{2} x^{2} + . . .^{n} C_{n} x^{n}] - n x - 1$
$=^{n} C_{2} x^{2} + . . .^{n} C_{n} x^{n}$
Hence it is divisible by $x$ and $x^{2}$

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