# PROBLEM 29

### S Carmody

Your friend is using Descartes's Rule of Signs to find the number of negative real roots of x^3 + x^2 + x + 1 = 0. Describe and correct the error

P(-x) = (-x)^3 +(-x)^2 + (-x) +1

= -x^3 - x^2 - x +1

"Because there is only on sign change in P(-x), there must be one negative real root"

### Work

The friend didn't do (-x)(-x)^2 correctly. They forgot that a negative times a negative is a positive. (Shown above in work)

### Conclusion!

There must be 3 or 1 negative roots.