Math III Section 4.6 #29 and 31

Ali'ce Batten

Problem 29

First evaluate the possible positive roots using ƒ(x) = x^3 + x^2 + x + 1

There are 0 sign change(s)

To find the remaining possible positive roots count down in pairs until you pass zero.

There are (0) possible combinations of positive roots

Calculate possible negative roots:

Given ƒ(x) = x^3 + x^2 + x + 1, first determine ƒ(-x)

ƒ(-x) = (-x)^3 + (-x)^2 + (-x) + 1

-x raised to an even power is positive. Odd exponents become negative:

(-x)^3 has a positive constant and odd exponent for a negative result of –x^3

(-x^)2 has a positive constant and even exponent for a positive result of + x^2

(-x) has a positive constant and odd exponent for a negative result of - x

1 has a positive constant and even exponent for a positive result of + 1

ƒ(-x) = -x^3 + x^2 - x + 1

First evaluate the possible negative roots using ƒ(x) = - x3 + x2 - x + 1

There are 3 sign change(s) detailed below:

Sign Change 1) - to +

Sign Change 2) + to -

Sign Change 3) - to +

To find the remaining possible negative roots, count down in pairs until you pass zero.

3 roots - 1 pair (2 roots) = 1

Therefore, you have a possible combination of (3 or 1) negative roots.

The Student

"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."

The mistake lies with the “-x^2”. A –x raised to an even exponent will result in a positive 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?

Please feel free to sketch a picture as I have chosen to do above.Since we are trying to fine the lengths of a trapezoid, I suggest you write down the formula.  Based on the problem we know that the area (A) is equal to sixty (A=60). The short base is equal to 2h(twice the height) and the longer base is equal to 2h+4 (the short base plus four). Now that the bases have been identified, plug in the expressions. Set the formula equal to 60 and solve for "h". After "h" has been found plug it into the original expressions and check you answer to make sure everything is right.