6.09 Polynomial Functions Activity

Materials Used: box of spaghetti, pencil and paper, Geogebra

Procedure:

1. Measure and record the length, width and height of the rectangular box you have chosen. Be sure to use the same measurement for all three dimensions (either centimeters or inches).

The rectangular box I chose to measure was a box of spaghetti. The length was 10 inches, the width was 3 inches, and the height was 2 inches.

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

V = lwh V = (10)(3)(2) V = 30(2) V = 60 inches squared

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).

V = lwh

60 = (x+0)(x-7)(x-8)

4. Simplify the equation and write it in standard form. If the equation contains decimals, multiply each term by a constant that will make all coefficients integers.

60 = (x+0)(x-7)(x-8)

f(x) = x^3 - 15x^2 + 56x -60

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.

Fundamental Theorem of Algebra:

The fundamental theorem of algebra states that the number of zeroes is equal to the degree of the polynomial.

F(x) = x^3 - 15x^2 + 56x -60

There are three zeroes.

Rational Root Theorem:

F(x) = x^3 - 15x^2 + 56x -60

In the rational root theorem, the constant term is represented by the letter p, and the leading coefficient is represented by...