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# Math III Section 4.6 Problems 29 & 31

## Problem 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."

When simplifying the equation "P(-x) = (-x)^3 +(-x)^2 + (-x) +1, (-x)^2 becomes (x)^2, because a positive times a positive equals a negative. This changes the answer. Instead of 1 sign change and 1 negative root, there is 3 sign changes and 3 or 1 negative root.

## Problem 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?

For the gardener to only have enough topsoil to cover 60 square feet of a trapezoid-shaped garden and wanting the shorter base to be twice the height, and the longer base to be 4 feet longer than the shorter base, the garden height would be 5 feet, a longer base of 14 feet, and a short base of 10 feet.