# Math III Sections 4.6 #29 and 31

### A. McKinnon

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 one sign change in P(-x), there must be one negative real root."

The original student was wrong because with the -x^2 he kept it negative but when a negatives squared it equals positive and this makes 2 sign changes which makes it a possibility of 2 or 0 negative solutions.

A gardener is designing a new garden in the shape of a trapezoid. She wants the shorter base to be twice the height and the longer base to be 4 feet longer than the shorter base. If she has enough topsoil to create a 60ft^2 garden, what dimensions should she use for the garden?

The height of the trapezoid is 5 feet, the shorter base of the trapezoid is 10 feet, and the longer base is 14 feet.