Question

Let p(x) be a polynomial with real coefficients such that the product of all six roots of p(x) is negative. Show that if the degree of p(x) is 6, then p(x) has at least one positive root.

Answer 1

Answer:

The product of 6 numbers is negative only is the number of negatives is odd.

If the number of negative roots is even, then their product will be positive.

The greatest odd number less than 6, is 5. It means the smallest number of positive numbers is 6 - 5 = 1.

So the least number of positive roots is one.