Title: Polynomial Functions
Materials used: * A cylindrical object such as a soup can or thermos * Ruler or tape measure * Graphing technology (e.g., graphing calculator or GeoGebra)
1. Measure and record the diameter and height of the cylindrical object you have chosen in inches. Round to the nearest whole number. 2. Apply the formula of a right circular cylinder (V = r2h) to find the volume of the object. (Note: Be sure to find the radius from the diameter measurement by dividing by 2.)
Now suppose you knew the volume of this object and the relation of the height to the radius, but did not know the radius. Rewriting the equation with one variable would result in a polynomial equation that you could solve to find the radius. 3. Rewrite the formula using the variable x for the radius. Substitute the value of the volume found in step 2 for V and express the 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 radius, then the expression for the height will be (x + 4). 4. Simplify the equation and write it in standard form. Multiply each term in the equation by 100 to eliminate any decimals, if necessary. 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.) 6. Substitute 0 for the function notation and, using graphing technology, graph the function. 7. Answer the following questions: * What does the Fundamental Theorem of Algebra indicate with respect to this equation? * What are the possible rational solutions of your equation? * How many possible positive, negative and