To address the problem described in the image:

We are tasked with finding:

1. A polynomial that represents the amount of space in the larger box that must be filled with Styrofoam chips.
2. Factoring the polynomial.

Step 1: Define Volumes of the Boxes

• Volume of the larger box: This is a cube with dimensions 12 inches on each side.

  The formula for the volume of a cube is: $V = \text{side}^3$ So, the volume of the larger box is: $V_{\text{larger}} = 12^3 = 1728 \text{ cubic inches.}$

• Volume of the smaller box: The smaller box has dimensions $x \times x \times x$.

  The volume of the smaller box is: $V_{\text{smaller}} = x^3 \text{ cubic inches.}$

Step 2: Amount of Space for Styrofoam Chips

The amount of space that must be filled with Styrofoam chips is the difference between the volumes of the larger and smaller boxes: $\text{Space for Styrofoam} = V_{\text{larger}} - V_{\text{smaller}} = 1728 - x^3$ Thus, the polynomial that describes the space for the Styrofoam chips is: $P(x) = 1728 - x^3$

Step 3: Factor the Polynomial

The polynomial $P(x) = 1728 - x^3$ is a difference of cubes. The difference of cubes can be factored using the formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ In this case, $a = 12$ and $b = x$. Applying the formula, we get: $P(x) = (12 - x)(12^2 + 12x + x^2)$ Simplifying further: $P(x) = (12 - x)(144 + 12x + x^2)$

Thus, the factored form of the polynomial is: $P(x) = (12 - x)(144 + 12x + x^2)$

Final Answer:

1. The polynomial that describes the amount of space to be filled with Styrofoam is $P(x) = 1728 - x^3$.
2. The factored form of the polynomial is $(12 - x)(144 + 12x + x^2)$.

Would you like any further explanation, or do you have any specific questions?

Relative Questions:

1. What is the general formula for the volume of a cube?
2. How do you calculate the difference between volumes of two objects?
3. What is the difference between factoring a polynomial and expanding it?
4. How is the difference of cubes factored in algebra?
5. How can factoring help in solving real-life packaging problems?

Tip:

Remember that recognizing special factoring patterns, like the difference of cubes, can simplify complex polynomial expressions.