1954 AHSME Problems/Problem 41

Problem 41

The sum of all the roots of $4x^3-8x^2-63x-9=0$ is:

$\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ -8 \qquad \textbf{(D)}\ -2 \qquad \textbf{(E)}\ 0$

Solution 1

By Vieta's Formulas, $\frac{--8}{4}=2$, $\fbox{B}$

Solution 2



By Vieta's Formulas: $x^3+ax^2+bx+c=(x-r)(x-p)(x-q)$



We can see that the negative sum of the roots is the coefficient of the $x^2$ term, $-2$. So the actual sum of the roots is $-(-2)$, or $2 \implies \fbox{B}$