Question

Eric multiplied (6x^2−7x+4.7)(x^2−2.8) using an area model.

Complete the blank spaces in the area model and determine the product in standard form.
6x^2 -7x 4.7
x^2 ___ -7x^3 ____
-2.8 ____ ____ -13.16

The product in standard form is

Answer 1

To fill in the area model and calculate the product of \((6x^2 - 7x + 4.7)(x^2 - 2.8)\), we'll compute each area of the rectangles in the model and then combine like terms.

### Setting Up the Area Model
The rows of the area model represent the first polynomial \( (6x^2 - 7x + 4.7) \) and the columns represent the second polynomial \( (x^2 - 2.8) \).

### Calculate Each Area
1. **For the top left rectangle:**
\[ 6x^2 \cdot x^2 = 6x^4 \]

2. **For the top middle rectangle:**
\[ 6x^2 \cdot (-2.8) = -16.8x^2 \]

3. **For the middle left rectangle:**
\[ -7x \cdot x^2 = -7x^3 \]

4. **For the middle middle rectangle:**
\[ -7x \cdot (-2.8) = 19.6x \]

5. **For the bottom left rectangle:**
\[ 4.7 \cdot x^2 = 4.7x^2 \]

6. **For the bottom middle rectangle:**
\[ 4.7 \cdot (-2.8) = -13.16 \]

### Filled Area Model
Now we fill in the computed areas into the area model:

```
| 6x^2 | -7x | 4.7
---------------------------------------
x^2 | 6x^4 | -7x^3 | 4.7x^2
---------------------------------------
-2.8 | -16.8x^2 | 19.6x | -13.16
```

### Combine Like Terms
Now we will combine all terms from the area model:
- **From \(6x^4\)**: \(6x^4\)
- **From \(-7x^3\)**: \(-7x^3\)
- **From the \(x^2\) terms**:
\[
-16.8x^2 + 4.7x^2 = -12.1x^2
\]
- **From the \(x\) terms**: \(19.6x\)
- **Constant term**: \(-13.16\)

### Final Expression in Standard Form
The product in standard form will be:

\[
6x^4 - 7x^3 - 12.1x^2 + 19.6x - 13.16
\]

### Conclusion
Thus, the completed area model and the product of the polynomials in standard form is:
\[
6x^4 - 7x^3 - 12.1x^2 + 19.6x - 13.16
\]