Question

A box is to be mailed. The volume in cubic inches of the box can be expressed as the product of its three dimensions: V(x) = x^3 - 16x^2 + 79x - 120. The length is (x - 8). Find the linear expressions for the width and the height. Assume that the width is greater than the height.

Answer 1

The answer is: w = x - 3, h = x - 5

V = l * w * h
V = x³ - 16x² + 79x - 120
l = x - 8

x³ - 16x² + 79x - 120 = (x - 8) * w * h
w * h = (x³ - 16x² + 79x - 120) / (x - 8)


                    x³ - 16x² + 79x - 120
x - 8 (* x²) |  x³ - 8x²
_________________________ (subtract)
                        -8x² + 79x - 120
   x - 8 (* -8x) | -8x² + 64x
_________________________ (subtract)
                                   15x - 120
                x - 8 (* 15) | 15x - 120
_________________________ (subtract)
                                          0

w * h = (x³ - 16x² + 79x - 120) / (x - 8) = x² - 8x + 15

Let's find factors of x² - 8x - 15:
x² - 8x - 15 = x² - 3x - 5x + 15
                  = x * x - 3 * x - (5 * x - 5 * 3) 
                  = x(x - 3) - 5(x - 3) 
                  = (x - 3)(x - 5)

w * h = (x - 3) * (x - 5)

w > h and (x - 3) > (x - 5)
⇒ w = x - 3
h = x - 5