Question

Let $f\left(x\right)=x^5+a{x}^4+b{x}^3+c{x}^2+dx-420$ where $a,b,c,d$ are real parameters, be a polynomial. If all zeros of the polynomial $f\left(x\right)$ are integers larger than $1,$ and $f\left(4\right)$ is equal to $k,$ then $k$ is divisible by

• A. $2$
• B. $3$
• C. $5$
• D. $6$

Answer 1

Answer: (A), (B), (D)

$6$

$f\left(x\right)=0$ has $5$ integral (not necessarily distinct) roots $d_1,d_2,...,d_5$
Then $f\left(x\right)=(x-d_1)(x-d_2)(x-d_3)(x-d_4)(x-d_5)$
Product of roots, $d_1{d}_2{d}_3{d}_4{d}_5=420$
and $420=2^2\cdot 3\cdot 5\cdot 7$

All of the roots are integers larger than $1,$ so they must be $2,2,3,5$ and $7.$
So $f\left(x\right)=(x-2{)}^2(x-3\left)\right(x-5\left)\right(x-7)$
Putting $x=4$ gives $k=12.$