Question

Eric multiplied (6x2−7x+4.7)(x2−2.8)
using an area model.

Complete the blank spaces in the area model and determine the product in standard form.

(5 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
6x2
−7x
4.7
x2
Response area −7x3
Response area
−2.8
Response area Response area −13.16
The product in standard form is
-19.6,6x3, -6x4,6x4-7x3-12.1x2+19.6x-13.16, 4.7x2,-7x3,19.6x,-16.8x.-7x2, -6x4-7x3-12.1-19.6+13.16,6x4+7x3-12x2-19x-13.16,6x3-7x2,12.1x+19.6+13.16,6x4,7x3

Answer 1

To multiply the expressions \((6x^2−7x+4.7)(x^2−2.8)\) using an area model, we can break it down into parts. First, we will identify the components of the expression.

The area model involves creating a box for each term in the first expression multiplied by each term in the second expression. We can outline it as follows:

### Breaking Down the Expressions
- First expression: \(6x^2, -7x, 4.7\)
- Second expression: \(x^2, -2.8\)

### Area Model Setup
1. **Top Row**:
- \(6x^2\)
- \(-7x\)
- \(4.7\)

2. **Side Column**:
- \(x^2\)
- \(-2.8\)

### Filling in the Affected Areas
- **Area for \(6x^2 \cdot x^2\)**:
- \(6x^4\)

- **Area for \(6x^2 \cdot -2.8\)**:
- \(-16.8x^2\)

- **Area for \(-7x \cdot x^2\)**:
- \(-7x^3\)

- **Area for \(-7x \cdot -2.8\)**:
- \(19.6x\)

- **Area for \(4.7 \cdot x^2\)**:
- \(4.7x^2\)

- **Area for \(4.7 \cdot -2.8\)**:
- \(-13.16\)

### Collecting Like Terms
Now we need to gather the terms:

1. \(6x^4\) (from the \(6x^2 \cdot x^2\))
2. Combine \(-7x^3\) (from \(-7x \cdot x^2\)) with terms from \(4.7 \cdot x^2 + -16.8x^2\):
- \(-7x^3\)
- \(4.7 - 16.8 = -12.1\) for \(x^2\)
3. \(19.6x\) (from \(-7x \cdot -2.8\))
4. Finally, combine the constant: \(-13.16\)

### The Standard Form of the Product
Arranging these in descending order of degree yields the product in standard form:

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

### Final Filling of the Response Areas
Using the areas derived:
- The response areas that needed to be filled would therefore include:
- **\(6x^4\)** for \(6x^2 \cdot x^2\)
- **\(-7x^3\)** for \(-7x \cdot x^2\)
- **\(-12.1x^2\)** for the aggregate \(x^2\) term
- **\(19.6x\)** for \(+19.6x\)
- **\(-13.16\)** for the constant

Thus, the final expression in standard form is:
\[
\boxed{6x^4 - 7x^3 - 12.1x^2 + 19.6x - 13.16}
\]