Suppose a box is to be constructed from a square piece of material of side length x by cutting out a 3-inch square from each corner and turning up the sides. Express the volume of the box as a polynomial in the variable x

material has base x inches by x inches

We are cutting out 3 inches at all 4 corners

base of box is (x-6) by (x-6)
height of box is 3

volume = 3(x-6)^2 , where x > 6

thank you

To find the volume of the box, we need to determine the dimensions of the box first.

When we cut out a 3-inch square from each corner of the original square piece of material, the resulting length of each side will be reduced by 3 inches.

Thus, the length and width of the base of the box will be (x - 6) inches. The height of the box will be the length of the material that was cut off, which is 3 inches.

Now, we can calculate the volume of the box by multiplying the length, width, and height:

Volume = (x - 6)(x - 6)(3)
= 3(x - 6)^2

Therefore, the volume of the box can be expressed as the polynomial 3(x - 6)^2.

To find the volume of the box, we need to determine the dimensions of the base and the height.

The base of the box will be a square created by cutting out squares with side length 3 inches from each corner of the original square. So, each side of the base will have a length of x - 2 * 3 inches, or x - 6 inches.

The height of the box will be equal to the length of the square pieces that were cut out, which is 3 inches.

Now, to calculate the volume of the box, we multiply the length, width, and height (base area * height). The volume can be expressed as:

Volume = (x - 6) * (x - 6) * 3

Simplifying this equation, we get:

Volume = 3(x - 6)^2

Therefore, the volume of the box can be expressed as the polynomial 3(x - 6)^2.