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# #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. #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?

### According to Decarte's rule of sign, the number of negative real roots of p(x)=0 is either equal to the number of sign changers between consecutive coefficients of p(-x) or is less than that by an even number.

"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."

• ** This is wrong because when she changed from P(x) to P(-x) she switched the x^2 to -x^2. It should have been positive and because it would have been positive you would have had 3 sign changes, not 1.**

To find possibilities for negative real roots, count the number of sign changes in the equation for p(-x).

Count the sign changes..

• because there is 3 which is and odd number, you use 3 and count by odds to zero...
• `so there is 3 or 1 negative real roots

# #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?

separate...

and now the next sentence with h=height...

now plug in the information

now solve for H..

so...

h=5 b1=14 b2=10