Math III Section 4.6 #29 

By: Noah Shrock

#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."

Below is my work shown for this problem

My friend's error is that there was three sign changes instead of one. Since there were three sign changes there was 1,3  negative real roots as my work shows.

#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?

Here is the brace map below.

Here is the the brace map for the work.

Below here is the work for the problem.

In conclusion, if a gardner only has enough topsoil for 60 square feet of a trapezoid garden, and they want the shorter base to be twice the height, and the longer base to be 4 feet longer than the shorter base; They should have a trapezoid gardner with a height of 5 feet, a shorter base 10 feet, and a longer base 14 feet.

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