Break it Down 1 Copyright © 2026 Pearson Education, Inc. Break it Down A packaging company specializes in designing custom boxes for everything from small electronics to food containers. The engineers use polynomials to represent dimensions, surface areas, and material costs. Each variable represents a measurable aspect, including length, width, and cost per unit. These expressions often include a base cost (a constant) and a variable cost (a term that changes based on size or quantity). To plan efficiently, the company frequently adds, subtracts, multiplies, and divides polynomials to simplify expressions, compare options, and calculate totals, which allows them to optimize production, reduce waste, and ensure that they stay on budget. 1. The surface areas of two box parts are modeled by 3x2 + 2x + 5 and 4x2 – x + 3. Write the expression for the sum of the surface areas. Represent your work visually by drawing a box model or use an algebra tiles diagram to show how the terms combine. 2. The total material used for two box designs is modeled by 6x2 + 5x + 9 and 2x2 + 3x + 4. Write the expression for the difference in total material between the two designs. Write a brief explanation or model to support your work. 3. A box face has dimensions x + 3 and x + 2. Find the area of the box face. 4. A rectangular box has base area modeled by the expression (2x + 3)(x + 5). a. Expand the expression to calculate the area in terms of x. b. If the height is x + 1, what expression models the volume of the box? c. Expand and simplify the volume.
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