Crayon box
Length: 3, Width: 1, Height: 4, Volume: 12
12 = (x) (x-2) (x+1)
Standard form: 12 = x3 - x2 - 2x
f(x) = x3 - x2 - 2x -12
I just want to know if I can write my equation: 12=(x+1)(x-2)(x)? Because if I don't include that last "x," then when I solve my equation it will only equal four. However, if I do add the "x," it will equal 12 but when I have to write it in standard form (ax^2+bx+c=0), it will be x^3-x^2-2x = 12. So am I supposed to write it like x^3-x^2-2x-12 = 0? Or like above (x^3-x^2-2x = 12). Either way, I have to factor it and I'm not sure how to do that. Thanks in advance
Option 2 - Rectangular Box
You will need the following materials to find the volume of a rectangular box:
A rectangular box such as a cereal or shoe box
Ruler or tape measure
Graphing technology (e.g., graphing calculator or GeoGebra)
Procedure:
Measure and record the length, width and height of the rectangular box you have chosen in inches. Round to the nearest whole number.
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.
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).
Simplify the equation and write it in standard form.
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.
