Let \(k\) and \(m\) be real numbers, and suppose that the roots of the equation \(x^3 - 7x^2 + kx - m = 0\) are three distinct positive integers. Compute \(k+m\).
+6248 \(\text{Applying Vieta's equations we have}\\ -7 = -(r_1+r_2+r_3)\\ \text{As we are told the roots are distinct positive integers the only possible solution is}\\ r_1=1,~r_2=2,~r_3=4,~\text{or some permutation thereof}\\ \text{Again using Vieta's equations (which return the same answer given all permutations)}\\ k = 1\cdot 2 + 1\cdot 4 + 2\cdot 4 = 14\\ m = 1\cdot 2\cdot 4 = 8\\ k+m=22\)
.
