Question
show me a area model represents the expression (3x+3)(x+2)
Answers
GPT-4o mini
An area model is a great way to visualize the distributive property and understand how to multiply two binomials. To represent the expression \((3x + 3)(x + 2)\) using an area model, follow these steps:
1. **Set Up the Rectangles**: Create a rectangle divided into smaller rectangles based on the terms in each binomial.
2. **Label the Dimensions**:
- The first binomial \(3x + 3\) can be represented as the height of the rectangle.
- The second binomial \(x + 2\) can be represented as the width of the rectangle.
3. **Divide the Rectangle**: The larger rectangle will be divided into four smaller rectangles based on the distribution of terms:
- The dimensions of the four smaller rectangles will be:
- The first rectangle will be \(3x \cdot x\)
- The second rectangle will be \(3x \cdot 2\)
- The third rectangle will be \(3 \cdot x\)
- The fourth rectangle will be \(3 \cdot 2\)
4. **Calculate the Area of Each Small Rectangle**:
- For the rectangle with dimensions \(3x\) and \(x\):
\[
Area_1 = 3x \cdot x = 3x^2
\]
- For the rectangle with dimensions \(3x\) and \(2\):
\[
Area_2 = 3x \cdot 2 = 6x
\]
- For the rectangle with dimensions \(3\) and \(x\):
\[
Area_3 = 3 \cdot x = 3x
\]
- For the rectangle with dimensions \(3\) and \(2\):
\[
Area_4 = 3 \cdot 2 = 6
\]
5. **Combine the Areas**: Finally, add up the areas of the four smaller rectangles to find the total area, which represents the product:
\[
Total = 3x^2 + 6x + 3x + 6
\]
Combine like terms:
\[
Total = 3x^2 + 9x + 6
\]
So, using the area model, the expression \((3x + 3)(x + 2)\) is represented by the area \(3x^2 + 9x + 6\). This gives you both a visual representation and the final simplified expression.
### Visual Representation
If you were to draw the area model, it would look like this (not to scale):
```
+------------+------------+
| 3x^2 | 6x |
| (3x)(x) | (3x)(2) |
+------------+------------+
| 3x | 6 |
| (3)(x) | (3)(2) |
+------------+------------+
```
You can then sum the areas of each rectangle to find the total area, which represents the product.
1. **Set Up the Rectangles**: Create a rectangle divided into smaller rectangles based on the terms in each binomial.
2. **Label the Dimensions**:
- The first binomial \(3x + 3\) can be represented as the height of the rectangle.
- The second binomial \(x + 2\) can be represented as the width of the rectangle.
3. **Divide the Rectangle**: The larger rectangle will be divided into four smaller rectangles based on the distribution of terms:
- The dimensions of the four smaller rectangles will be:
- The first rectangle will be \(3x \cdot x\)
- The second rectangle will be \(3x \cdot 2\)
- The third rectangle will be \(3 \cdot x\)
- The fourth rectangle will be \(3 \cdot 2\)
4. **Calculate the Area of Each Small Rectangle**:
- For the rectangle with dimensions \(3x\) and \(x\):
\[
Area_1 = 3x \cdot x = 3x^2
\]
- For the rectangle with dimensions \(3x\) and \(2\):
\[
Area_2 = 3x \cdot 2 = 6x
\]
- For the rectangle with dimensions \(3\) and \(x\):
\[
Area_3 = 3 \cdot x = 3x
\]
- For the rectangle with dimensions \(3\) and \(2\):
\[
Area_4 = 3 \cdot 2 = 6
\]
5. **Combine the Areas**: Finally, add up the areas of the four smaller rectangles to find the total area, which represents the product:
\[
Total = 3x^2 + 6x + 3x + 6
\]
Combine like terms:
\[
Total = 3x^2 + 9x + 6
\]
So, using the area model, the expression \((3x + 3)(x + 2)\) is represented by the area \(3x^2 + 9x + 6\). This gives you both a visual representation and the final simplified expression.
### Visual Representation
If you were to draw the area model, it would look like this (not to scale):
```
+------------+------------+
| 3x^2 | 6x |
| (3x)(x) | (3x)(2) |
+------------+------------+
| 3x | 6 |
| (3)(x) | (3)(2) |
+------------+------------+
```
You can then sum the areas of each rectangle to find the total area, which represents the product.