Math III Section 4.6 #29 & #31

J. Bell Mr. Kirkland's Class

Problem # 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 on sign change in P(-x), there must be one negative real root."

Show the correct work on your paper, take a picture, and add it to your Tackk. Use complete sentences in text boxes that describe the error and describe the correction.

First we're going to want to set up our problem...

Now that we've set up our equation in a simple and easy to see format. We can change the problems exponents to negatives so that we may find out how many possible negative roots are present.

Now lets work it out...

Once we've set up all of our exponents to be negative we can work it out and figure out what exactly is going to be changed and what is going to be kept the same.

Once we've figured out and combined all of our exponents once again, we quickly notice that this is where the student in question made his mistake by confusing his signs and negative exponents.

By looking out the now simplified equation we can example the different sign changes. As stated above we find that there are 3 sign changes in P(-x) so with this information we can deduct that there must be either 3 or 1 negative roots to go with this equation.

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?

Use a Brace Map to take the whole word problem and separate it into its parts. The area of a trapezoid is A = (1/2)(h)(b1 + b2). Take pictures or screen shots of your work and add them to your Tackk. Use text boxes and complete sentences to explain your work and answer the question.

Setting up our problem!

First we're going to want to set up our problem by using the formula for finding out the area of a trapezoid which is: A=1/2 h (b1 + b2). And by looking at the word problem we can dissect from it that our numbers are as follows...

Now to work it out…

By taking the above information we can set our problem up into our formula using the corresponding numbers: A=60 h= unknown to us which is what we are trying to find and the w=2h and l=2h+4. The way we would set it up and work it out is shown below.

Once we've gotten our problem simplified to this point we can now attempt to solve for h. Usually we would use the factoring method to find h but since we quickly find out that there is no way to factor we are instead going to use the quadratic formula which is -b + or - the square root of b^2-4ac all over 2a. It is shown below how you would plug the numbers above into the formula.

After simplifying to the point where we either + or - (-2) with 22 we can do both ways to find two numbers which are 5 and 6. Then we use these two numbers to plug into the original equation so that we can see which one we are going to use for h.

After working out the problem we find that 5 works in the equation to equal the exact number 60 which is our limit. And we may conclude that 5 is equal to our height and that is our answer.

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