The sum of two of the three roots of x^3 + ax^2 + bx + c = 0 is 0. Write an equation expressing c explicitly in terms of a and b.
+26367 The sum of two of the three roots of \(x^3 + ax^2 + bx + c = 0 \) is \(0\).
Write an equation expressing c explicitly in terms of \(a\) and \(b\).
Let the roots are \(x_1,\) \(x_2\) and \(x_3\).
\(\begin{array}{|rcll|} \hline x^3 + ax^2 + bx + c &=& (x-x_1)(x-x_2)(x-x_3) = 0 \quad | \quad x_1+x_2 = 0\ \text{ or }\ x_2 = -x_1 \\\\ &=& (x-x_1)\Big(x-(-x_1)\Big)(x-x_3) \\ &=& (x-x_1)(x+x_1)(x-x_3) \\ &=& (x^2-x^2_1)(x-x_3) \\ x^3 + ax^2 + bx + c &=& x^3-x_3x^2-x^2_1x+x^2_1x_3 \\ && \text{compare} \\ && \boxed{a=-x_3 \text{ or } x_3=-a \\ b=-x^2_1 \text{ or } x^2_1=-b \\ c= x^2_1x_3 \text{ or } c=(-b)(-a)=ab } \\ \hline \end{array}\)
\(\boxed{c=ab} \)
