# Section 4.6 #29 and #31

### J. Jones

**(29.)"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."

Begin by finding positive roots with the problem ƒ(x) = x^3 + x^2 + x + 1. There are NO sign changes. Therefore there are (0) possible combinations of positive roots. Calculate the possible negative roots:

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

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

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

**The error was that the student should have kept x^2 positive rather than make it negative with the other changes in (x).**

**(31.) 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?**

Brace Map

1. A gardner is designing a new garden in the shape of a trapezoid.

2. She wants the shorter base to be twice the height and the longer base to be 4 feet longer than the shorter base.

3. If she has enough topsoil to create a 60ft^2 garden, what dimensions should she use for the garden?

As stated above, the results of the work end with the Height being 5ft, B1 (larger base) being 14 ft, and B2 (smaller base) being 10 ft.