Let's dissect the given expression [tex]\((3x + 1)(x - 1)\)[/tex] and determine what it represents by expanding it step-by-step. This will help us understand the underlying geometrical or numerical representation it might correspond to.
To expand the expression [tex]\((3x + 1)(x - 1)\)[/tex], we can use the distributive property (also known as the FOIL method for binomials) to multiply it out.
1. First, distribute [tex]\(3x\)[/tex] to both terms in the second binomial:
[tex]\[
3x \cdot (x - 1) = 3x \cdot x - 3x \cdot 1 = 3x^2 - 3x
\][/tex]
2. Next, distribute [tex]\(1\)[/tex] to both terms in the second binomial:
[tex]\[
1 \cdot (x - 1) = 1 \cdot x - 1 \cdot 1 = x - 1
\][/tex]
3. Combine these results:
[tex]\[
3x^2 - 3x + x - 1
\][/tex]
4. Combine like terms:
[tex]\[
3x^2 - 2x - 1
\][/tex]
So, the expanded form of the expression [tex]\((3x + 1)(x - 1)\)[/tex] is:
[tex]\[
3x^2 - 2x - 1
\][/tex]
Given this polynomial [tex]\(3x^2 - 2x - 1\)[/tex], let's interpret what it might represent within the context of a floor plan of a house where dimensions are given in terms of [tex]\(x\)[/tex].
This polynomial could represent an area, since area is typically characterized by quadratic expressions when dimensions are given in terms of linear measurements.
Therefore, one possible scenario is that the expression [tex]\((3x + 1)(x - 1)\)[/tex] represents the area of a specific part of the house, such as a room or a combination of rooms, where:
- [tex]\(3x + 1\)[/tex] and [tex]\(x - 1\)[/tex] are linear dimensions (length and width) of a rectangular area.
- Multiplying these dimensions gives us the quadratic [tex]\(3x^2 - 2x - 1\)[/tex], which is the area of that rectangular space within the house's floor plan, represented in square units.
In summary, the expression [tex]\((3x + 1)(x - 1)\)[/tex] can be interpreted as representing the area of a rectangular section of the house where one dimension is [tex]\(3x + 1\)[/tex] units and the other dimension is [tex]\(x - 1\)[/tex] units.
