Math III Section 4.6 #29 and #31

"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."

Problem #29

When my friend simplified the problem as P(-x)= -x^3 -x^2-x+1, they forgot that when the negative (x) is squared, it becomes a positive (x), because a negative times a negative equals a positive. Although they forgot that one minor detail, it makes a huge difference in the answer. My friend stated that with his P(-x) there was only 1 sign change, so one negative root. The correct answer is that there were 3 sign changes, and  there are 3 or 1 negative real roots.

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?

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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