Question

If $f(x)$ is a monic polynomial function of degree 4 satisfying $f(i)=\frac{1}{i}$ for $i=1,2,3,4$ then

• A. number of zeroes at the end of f (5)! is 4.
• B. number of divisors of f (5) is 8.
• C. sum of even divisors of f (5) is 56.
• D. sum of odd divisors of f (5) is 18.

Answer 1

Answer: (A), (B), (C)

$xf(x)-1\equiv (x-1)(x-2)(x-3)(x-4)(x-\alpha )$
Put $x=0\Rightarrow \alpha =\frac{1}{24}$
$\therefore f(x)=\frac{(x-1)(x-2)(x-3)(x-4)\left(x-\frac{1}{24}\right)+1}{x}$
$\Rightarrow f(5)=\frac{4\times 3\times 2\times 1\times \frac{119}{24}+1}{5}=24$
Now verify the options