Alright, let's factor the polynomial [tex]\(3x^2 + 4x + 1\)[/tex] using a geometric model step-by-step.
### Step 1: Model the Trinomial with Tiles
We will model the trinomial by representing each term with geometric tiles. Let's consider:
- [tex]\(3x^2\)[/tex] as a large square representing an area of [tex]\(3x^2\)[/tex].
- [tex]\(4x\)[/tex] as smaller rectangular tiles, each representing an area of [tex]\(x\)[/tex].
- [tex]\(1\)[/tex] as a small square tile representing an area of [tex]\(1\)[/tex].
### Step 2: Arrange Tiles to Form a Rectangle
To factor the trinomial, we aim to arrange these tiles into a single rectangle.
#### Place [tex]\(3x^2\)[/tex] Tiles
Position the [tex]\(3x^2\)[/tex] tiles (representation of three large square tiles, each [tex]\(x \times x\)[/tex]) in the top-left corner:
#### Place [tex]\(4x\)[/tex] Tiles
Next, we need to arrange the [tex]\(4x\)[/tex] tiles to fit around the [tex]\(3x^2\)[/tex] tiles. This can be done by placing them to the right and below the [tex]\(3x^2\)[/tex] tiles. We can experiment with different placements to find a perfect rectangle arrangement.
Place two [tex]\(x\)[/tex] tiles horizontally on the right of the [tex]\(3x^2\)[/tex] tiles.
Place two [tex]\(x\)[/tex] tiles vertically below the [tex]\(3x^2\)[/tex] tiles.
The arrangement forms a nearly completed large rectangle with the [tex]\(3x^2\)[/tex] tiles in the corner and the [tex]\(4x\)[/tex] tiles along its sides.
#### Place the [tex]\(1\)[/tex] Tile
Finally, the [tex]\(1\)[/tex] tile (a small square representing [tex]\(1\)[/tex]) fits perfectly into the corner that completes the rectangle.
### Step 3: Verify the Rectangle Formation
The geometric tiles should now form a perfect rectangle. We will now label the dimensions of this rectangle to identify our factors.
#### Label the Dimensions
The [tex]\(3x^2\)[/tex] tile provides the length that is a part of both dimensions. Looking at placements:
- Vertically aligned tiles ([tex]\(3x^2\)[/tex] and additional [tex]\(2x\)[/tex]), shows one dimension is [tex]\(x + 1\)[/tex].
- Horizontally aligned tiles ([tex]\(3x^2\)[/tex] with [tex]\(2x\)[/tex]), indicates the other dimension is [tex]\(3x + 1\)[/tex].
### Step 4: Identify the Factors
Hence, the dimensions of the rectangle, which represent the factors of the polynomial, are [tex]\(x + 1\)[/tex] and [tex]\(3x + 1\)[/tex].
### Conclusion
By arranging the tiles geometrically, we can determine that the factors of the trinomial [tex]\(3x^2 + 4x + 1\)[/tex] are:
[tex]\[
(3x + 1) \cdot (x + 1)
\][/tex]
Thus, the trinomial [tex]\(3x^2 + 4x + 1\)[/tex] can be factored into:
[tex]\[
3x^2 + 4x + 1 = (3x + 1)(x + 1)
\][/tex]
