Math III Section 4.6 #29 and #31

B. Vences

#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."

(Incorrect roots and answer)

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

= -x^3 - x^2 - x +1"

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

( Correct roots and reason why )

The error that occurred was when the substitution of P(-x) in the original equation was that +(-x)^2 = -x^2 is incorrect because a (-x)(-x) = + (x). So it is concluded that the correction to this error is x^2.

since the original equation was set up, then it can be assumed that the answer is wrong as well in that there isn't 1 negative real root, but 3. the first term, and second term shows different signs and according to descartes rule of sign changes, it counts as a negative real root. So if you were to compare the sign changes starting from the beginning of the equation to the very end, you get 3 sign changes. So the correct is "3 of less than the number".

#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?

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