Q.If $\alpha,\beta,\gamma,$ and $\delta$ are the roots of the polynomial equation $2x^4+5x^3-7x^2+8=0$, find a quadratic equation with integer coefficients whose roots are $\alpha+\beta+\gamma+\delta$ and $\alpha\beta\gamma\delta$.

Step 1. Read off the needed Vieta sums. For $2x^4+5x^3-7x^2+0x+8=0$: $\Sigma_1=-\dfrac52$ and $\Sigma_4=\dfrac82=4$.

Step 2. These two numbers ARE the required new roots. The new quadratic's roots are $r_1=\Sigma_1=-\tfrac52$ and $r_2=\Sigma_4=4$. …

Forming an Equation from Given Roots — the two required roots come directly fro …

• Confusing $\Sigma_1$ (sum of roots) with the coefficient of $x^3$ itself (a sign flip is needed) …