Problem 102 - Introductory Algebra for College Students

Question

A box with no top is to be made from an 8-inch by 6 -inch piece of metal by cutting identical squares from each corner and turning up the sides (see the figure). The volume of the box is modeled by the polynomial \(4 x^{3}-28 x^{2}+48 x .\) Factor the polynomial completely. Then use the dimensions given on the box and show that its volume is equivalent to the factorization that you obtain.

Step-by-Step Solution

Answer

The polynomial \(4x^3 - 28x^2 + 48x\), which represents the volume of the box, factors to \(4x(x - 4)(x - 3)\). This factorization aligns with the box's dimensions when squares of side x are cut from each corner, showing that the box's volume is consistent with the factored expression.

Step 1: Factorization of the polynomial

Firstly, factor the polynomial \(4x^3 - 28x^2 + 48x\) by looking for the common factors. Here, 4x is the greatest common factor (GCF) such that: \( 4x (x^2 - 7x + 12) \) is obtained.

Step 2: Factorization of the resulting polynomial

Again, factorize the quadratic equation \(x^2 - 7x + 12\). This can be factored into: \( (x - 4) (x - 3)\). Hence, the complete factorization of the volume expression becomes \(4x(x - 4)(x - 3)\).

Step 3: Verifying the factorization with given box dimensions

The metal sheet is cut into squares from each corner with side x. After the squares are cut off, the resulting fold-up sides produce a box with dimensions: height=x, length=(8-2x), and width=(6-2x). Hence, its volume is x*(8-2x)*(6-2x). Expand this expression, simplify it, and compare it with the initially given volume expression, which is \(4x^3 - 28x^2 + 48x\). If true, this validates that the factorization is correct.

Key Concepts

Volume of a Box

When constructing a box, especially one without a top, understanding how to calculate the volume is crucial. The volume of a box is determined by multiplying its three dimensions: length, width, and height. In mathematical terms, it is expressed as:
\[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \]
In the case of our exercise, the box's height is represented by \(x\), whereas the length and width are \(8 - 2x\) and \(6 - 2x\), respectively.

• Height (after folding) = \(x\)
• Length (after folding calculation) = \(8 - 2x\)
• Width (after folding calculation) = \(6 - 2x\)

These expressions arise from removing \(x\) from each dimension due to the squares cut from corners. So the box's volume is the product of these dimensions, \(x \times (8 - 2x) \times (6 - 2x)\). By understanding this, we can dissect the polynomial modeling the volume and verify its correctness.

Greatest Common Factor

In polynomial factorization, the greatest common factor (GCF) is a critical concept. A GCF is the highest factor that can evenly divide each term of the polynomial. It simplifies and breaks down complex polynomial expressions into manageable parts.
To identify the GCF in our exercise's initial polynomial, \(4x^3 - 28x^2 + 48x\), you compare each term's coefficients and variables:

• Coefficient Analysis: Maximum number dividing 4, 28, and 48 is 4.
• Variable Analysis: The lowest power of \(x\) across all terms is \(x\).

Thus, the GCF here is \(4x\). Factoring out \(4x\) from all terms results in a simplified \(4x(x^2 - 7x + 12)\). Recognizing this factor can quicken solving and helps in the polynomial's next level of factorization.

Quadratic Equation Factorization

Quadratic equation factorization is a process of expressing a quadratic as a product of its linear factors. Given a standard quadratic of the form \(ax^2 + bx + c\), we aim to express it as \((x - p)(x - q)\). These \(p\) and \(q\) are the roots, satisfying the equation.
In the exercise, we deal with the quadratic \(x^2 - 7x + 12\). The steps to factor this include:

• Identify two numbers multiplying to box term 12 (constant) but adding to -7 (middle coefficient).
• Here, the numbers are -3 and -4, because \(-3 \times -4 = 12\) and \(-3 + -4 = -7\).

So, \(x^2 - 7x + 12\) factors to \((x - 3)(x - 4)\). This final factorization step leads to the complete polynomial factorization: \(4x(x - 4)(x - 3)\), critical for simplifying expression and solving equations.