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Q.
If all the roots (zeros) of the polynomial $f(x)=x^5+a^4+b^3+ cx ^2+d x-420$ are integers larger than 1 , then $f (4)$ equals
Complex Numbers and Quadratic Equations
Solution:
$f$ must have 5 (not necessarily distinct) roots $d _1, d _2, \ldots \ldots . d _5, f$ 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.