jmc

Question

Let k and m be real numbers, and suppose that the roots of the equation x^3 - 7x^2 + kx - m = 0are three distinct positive integers. Compute k + m.

Accepted Answer

By Vieta's formulas, the sum of the roots of the equation is

7.

Furthermore, the only triple of distinct positive integers with a sum of

7

is

{1, 2, 4} .

To see this, note that the largest possible value for any of the three integers is

7 - 1 - 2 = 4,

and the only way to choose three of the integers

1, 2, 3, 4

to sum to

7

is to choose

1,

2,

and

4.

Therefore, the roots of the equation must be

1,

2,

and

4.

It follows by Vieta that

k = 1 \cdot 2 + 2 \cdot 4 + 1 \cdot 4 = 14

and

m = 1 \cdot 2 \cdot 4 = 8,

so

k + m = 14 + 8 = 22 .