Mathematics - Quadratic Equation Question with Solution | TestHub

Let ${\alpha }_{i}i=1,2,3,4$ the 4 integral roots of $x^n+a_1{x}^{n-1}+\dots +a_n=5$ and let K be an integral root of $x^n+a_1{x}^{n-1}+..+a_n=7$$\Rightarrow \left(x-{\alpha }_1\right)\left(x-{\alpha }_2\right)\left(x-{\alpha }_3\right)\left(x-{\alpha }_4\right)=2$ has an integral root K . $\Rightarrow \left(K-{\alpha }_1\right)\left(K-{\alpha }_2\right)\left(K-{\alpha }_3\right)\left(K-{\alpha }_4\right)=2$ $K-{\alpha }_i,\mathrm{i}=1,2,3,4$ are all integers and are distinct which is impossible ( ∵ product of 4 district integers cannot be 2 ). Hence $x^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\dots +a_n=7$ has no integral roots.