**6.09 Option 2**

**Justin Chacko**

5. Find the solutions to this equation algebraically using the Fundamental Theorem of Algebra, the Rational Root Theorem, Descartes' Rule of Signs, and the Factor Theorem.

Hint

If the numbers are large, graph the function first using GeoGebra to help you find one of the zeros. Use that zero to find the depressed equation which can be solved by factoring or the quadratic formula.

4. Simplify the equation and write it in standard form.

168 = (x + 1)(x - 4)(x)

168 = (x^2 - 4x + 1x - 3)(x)

168 = (x^2 - 3x - 4)(x)

V = x^3 - 3x^2 - 4x - 168

Step 2

2.Apply the formula of a rectangular box (V = lwh) to find the volume of the object. V = 168

Now suppose you knew the volume of this object and the relation of the length to the width and height, but did not know the length. Rewriting the equation with one variable would result in a polynomial equation that you could solve to find the length.

Step 5

Step 4

Procedure

Step 1

1. Measure and record the length, width and height of the rectangular box you have chosen in inches. Round to the nearest whole number.

Length: 7

Width: 3

Height: 8

Step 3

3. Rewrite the formula using the variable x for the length. Substitute the value of the volume found in step 2 for V and express the width and height of the object in terms of x plus or minus a constant. For example, if the height measurement is 4 inches longer than the length, then the expression for the height will be (x + 4). 168 = (x + 1)(x - 4)(x)

Step 6

6.Substitute 0 for the function notation and, using graphing technology, graph the function.

Step 7

7. Answer the following questions :

*What does the Fundamental Theorem of Algebra indicate with respect to this equation?

-According to the Fundamental Theorem of Algebra, this equation has 3 possible solutions because it is raised to the third degree.

*What are the possible rational solutions of your equation?

-The possible solutions were positive or negative of each of the following: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, and 168.

*How many possible positive, negative and complex solutions are there in your equation?

-There was 1 possible positive real, 2 or 0 possible negative real, and 2 or 0 possible complex.

*Graph the function. What type of function has been graphed (linear, quadratic, cubic, or quartic)? Provide your reasoning and describe the end behavior of the graph.

-This was a cubic type of function. Volume is cubic, and we were finding the volume of a rectangular box, so it makes sense that this was a cubic function. The left side continues downward and the right end continues upward.

*How do the solutions of the equation compare to the length of the rectangular object, and the x-intercept of the graph? Provide both the solutions and measurement.

-The solution was the same as the length measurement. The solution was 7, and the measurements were 7, 3, and 8.

Factors of (p/q)-

Positive or Negative of each of the following: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, and 168.

There is 1 sign change in the original function, so there is one positive real zero. There are 2 or 0 negative real zeros and 2 or 0 complex zeros.

If I use synthetic division, I can see that 7 is a zero of this function. This is the only real zero. If I tried to continue to find other zeros using the quadratic formula, I'd see that they are just imaginary numbers. (2 +/- 4.47i).

The object I used was an old wooden jewelry box. The materials I used were a ruler, the cox, and a graphing calculator online.

What did you think of the project? The project was pretty cool.

What did you learn? I learned how to make a function and graph it using the dimensions from a box.

Do you have any questions or concerns? Not really. I just hope I did this right.