To factor the quadratic polynomial [tex]\(2x^2 - x - 1\)[/tex] using a geometric model, we will follow a series of steps. Here's the detailed, step-by-step solution:
### Step 1: Initial Placement of Tiles
1. Model the polynomial by placing tiles in the Product section.
- We start by representing each term of the given polynomial [tex]\(2x^2 - x - 1\)[/tex] with tiles:
- [tex]\(2x^2\)[/tex]: Place two [tex]\(x^2\)[/tex] tiles in the Product section.
- [tex]\(-x\)[/tex]: Place one [tex]\(-x\)[/tex] tile, splitting it into two pieces of [tex]\(-\frac{x}{2}\)[/tex] each to facilitate arrangement later.
- [tex]\(-1\)[/tex]: Place one [tex]\(-1\)[/tex] tile in the Product section.
### Step 2: Arrange the Tiles to Form a Rectangle
2. Arrange the tiles in such a way that they form a rectangle.
- Combine and adjust the tiles to form a rectangle. This process helps in visualizing the factors of the polynomial.
Let’s map the pieces to form the rectangle:
- Arrange the two [tex]\(x^2\)[/tex] tiles side by side.
- Place the [tex]\(-\frac{x}{2}\)[/tex] tiles such that they share sides with the [tex]\(x^2\)[/tex] tiles.
- Finally, place the [tex]\(-1\)[/tex] tile to complete the rectangle.
### Step 3: Identify the Dimensions of the Rectangle
3. Identify the dimensions (factors of the polynomial) from the arrangement:
- Check the dimensions by examining the way the tiles are arranged.
- One dimension can be identified by combining the length of one [tex]\(x^2\)[/tex] tile and one dimension of the [tex]\(-\frac{x}{2}\)[/tex] pieces.
- The remaining dimension can be found by the other sides of the arranged tiles that fit together to form the rectangle.
### Step 4: Write the Polynomial in Factored Form
4. Convert the dimensions identified in Step 3 into algebraic expressions to write the polynomial in factored form:
- From the arrangement, we have identified two dimensions as:
- [tex]\((x - 1)\)[/tex]
- [tex]\((2x + 1)\)[/tex]
Thus, the quadratic polynomial factors into:
[tex]\[ (x - 1)(2x + 1) \][/tex]
### Conclusion
Therefore, the polynomial [tex]\(2x^2 - x - 1\)[/tex] factors into:
[tex]\[ (x - 1)(2x + 1) \][/tex]
This is the factored form of the quadratic polynomial.
