Question

If all the roots (zeros) of the polynomial $f(x)=x^5+a^4+b^3+c{x}^2+dx-420$ are integers larger than 1 , then $f(4)$ equals

• A. 0
• B. 12
• C. -6
• D. -12

Answer 1

Answer: (B)

$f$ must have 5 (not necessarily distinct) roots $d_1,d_2,\dots \dots .d_5,f$ factors as $\left(x-d_1\right)\left(x-d_2\right)\left(x-d_3\right)$ $\left(x-d_4\right)\left(x-d_5\right)$. The product $d_1{d}_2{d}_3{d}_4{d}_5$ must be equal to 420 , which factors as $2^2\cdot 3\cdot 5\cdot 7$. All of the roots are integers larger than 1 , so they must be $2,2,3,5$ and 7 . So $f(x)=(x-2{)}^2(x-3)(x-5)(x-7)$. Putting in $x=4$ gives 12.