3. Mrs. LeBlanc is making a Pokémon box. The length of the box is 12 inches greater than its width, and its height is 5 inches less than its width. Find the dimensions of the Pokémon box formed, given that its volume is 9600 cubic inches. (Show all work including setting up the equation, simplifying the equation, listing the possible rational roots, testing the possible rational roots using synthetic division, and then identifying all of the dimensions.

Let x inches be the width of the box. If the length of the box is 12 inches greater than its width, then the length of the box is x+12 inches. If the height of the box is 5 inches less than its width, then the height of the box is x-5 inches.

The volume of box is

[tex]V=\text{length}\cdot \text{width}\cdot \text{height}\ in^3.[/tex]

Therefore,

[tex]9600=x\cdot (x+12)\cdot (x-5).[/tex]

Solve this equation:

[tex]x(x^2+12x-5x-60)=9600,\\ \\x^3+7x^2-60x-9600=0.[/tex]

The integer solutions of this equation can be among the divisors of -9600. The divisors of 9600 are:

[tex]\pm1, \pm2, \pm3, \pm4,\pm5,\pm6,\pm8,\pm10,\pm12,\pm15,\pm16,\pm20,\pm24,\pm25,\pm30,\pm32,....[/tex]

We can check all these divisors and see that x=20 is a solution, then using synthetic division

[tex]x^3+7x^2-60x-9600=(x-20)(x^2+27x+480).[/tex]

Since the quadratic factor has discriminant [tex]D=27^2-4\cdot 480=-1191<0,[/tex] we can state that the only solution is x=20 inches. The width is 20 inches, the length is 32 inches and the height is 15 inches.