Algebra worksheet focusing on factoring common factors using area models.

Let's solve this step-by-step, using the concept of factoring a common factor using area models. The idea is to represent algebraic expressions as areas of rectangles, where one side is the common factor, and the other side is the sum of terms.

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Part 1: Fill in the missing information for each rectangle

We'll use the area model:
- Area = length × width
- If we have two parts of the rectangle, the total area is the sum of the areas of the parts.
- We factor out the greatest common factor (GCF) from both terms.

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#### 1.
```
x 6
┌─────┬─────┐
2 │ 2x │ 12 │
└─────┴─────┘
```

- Dimensions: `2` and `(x + 6)`
- Area as product: `2(x + 6)`
- Area as sum: `2x + 12`

✔ Already filled correctly.

---

#### 2.
```
□ □
┌─────┬─────┐
5 │ 5x │ 20 │
└─────┴─────┘
```

- One dimension is `5`, and the areas are `5x` and `20`.
- So, the top labels must be such that:
- `5 × ? = 5x` → `? = x`
- `5 × ? = 20` → `? = 4`
- So top: `x` and `4`
- Dimensions: `5` and `(x + 4)`
- Area as product: `5(x + 4)`
- Area as sum: `5x + 20`

✔ Answer:
- Top: `x`, `4`
- Dimensions: `5` and `(x + 4)`
- Product: `5(x + 4)`
- Sum: `5x + 20`

---

#### 3.
```
□ 8
┌─────┬─────┐
□ │ 6x │ 48 │
└─────┴─────┘
```

- Areas: `6x` and `48`
- Let’s find the GCF of `6x` and `48`. GCF = `6`
- So, one side is `6`, and the other side is `(x + 8)` because:
- `6 × x = 6x`
- `6 × 8 = 48`
- So:
- Left side: `6`
- Top left: `x`, top right: `8` (already given)
- Dimensions: `6` and `(x + 8)`
- Product: `6(x + 8)`
- Sum: `6x + 48`

✔ Answer:
- Left side: `6`
- Top left: `x` (missing), but already implied
- Dimensions: `6` and `(x + 8)`
- Product: `6(x + 8)`
- Sum: `6x + 48`

---

#### 4.
```
x □
┌─────┬─────┐
□ │10x │ 30 │
└─────┴─────┘
```

- Areas: `10x` and `30`
- GCF of `10x` and `30` is `10`
- So one side is `10`
- Then:
- `10 × x = 10x` → correct
- `10 × 3 = 30` → so top right must be `3`
- Left side: `10`
- Top: `x` and `3`
- Dimensions: `10` and `(x + 3)`
- Product: `10(x + 3)`
- Sum: `10x + 30`

✔ Answer:
- Left side: `10`
- Top right: `3`
- Dimensions: `10` and `(x + 3)`
- Product: `10(x + 3)`
- Sum: `10x + 30`

---

Part 2: Fill in the missing dimensions from the expression given

We now use the same area model idea.

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#### 5. $ 5x + 35 = 5( \_\_\_ ) $

- Factor out 5: $ 5x + 35 = 5(x + 7) $
- So blank: `x + 7`
- Rectangle:
```
x 7
┌─────┬─────┐
5 │ 5x │ 35 │
└─────┴─────┘
```

✔ Answer: `x + 7`

---

#### 6. $ 2x + 12 = 2( \_\_\_ ) $

- $ 2x + 12 = 2(x + 6) $
- Blank: `x + 6`
- Rectangle:
```
x 6
┌─────┬─────┐
2 │ 2x │ 12 │
└─────┴─────┘
```

✔ Answer: `x + 6`

---

#### 7. $ 3x - 21 = ( \_\_\_ ) $

- GCF is 3 → $ 3x - 21 = 3(x - 7) $
- Blank: `3(x - 7)`
- Rectangle:
```
x -7
┌─────┬─────┐
3 │ 3x │ -21 │
└─────┴─────┘
```

✔ Answer: `3(x - 7)`

---

#### 8. $ 7x - 21 = ( \_\_\_ ) $

- GCF is 7 → $ 7x - 21 = 7(x - 3) $
- Blank: `7(x - 3)`
- Rectangle:
```
x -3
┌─────┬─────┐
7 │ 7x │ -21 │
└─────┴─────┘
```

✔ Answer: `7(x - 3)`

---

#### 9. $ -3x - 15 = -3( \_\_\_ ) $

- Factor out `-3`:
$ -3x - 15 = -3(x + 5) $
- Because: $-3 \times x = -3x$, $-3 \times 5 = -15$
- Blank: `x + 5`
- Rectangle:
```
x 5
┌─────┬─────┐
-3 │-3x │ -15 │
└─────┴─────┘
```

✔ Answer: `x + 5`

---

#### 10. $ -5x + 45 = ( \_\_\_ ) $

- GCF is `-5` or `5`? Let's see:
$-5x + 45 = -5(x - 9)$
- Because: $-5 \times x = -5x$, $-5 \times (-9) = +45$
- So: $-5(x - 9)$
- Blank: `-5(x - 9)`
- Rectangle:
```
x -9
┌─────┬─────┐
-5 │-5x │ 45 │
└─────┴─────┘
```

✔ Answer: `-5(x - 9)`

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Part 3: Factor these

Now factor completely:

#### 11. $ 4x - 16 $

- GCF = 4 → $ 4(x - 4) $

✔ Answer: $ 4(x - 4) $

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#### 12. $ -7x - 35 $

- GCF = -7 → $ -7(x + 5) $

✔ Answer: $ -7(x + 5) $

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#### 13. $ 9x - 81 $

- GCF = 9 → $ 9(x - 9) $

✔ Answer: $ 9(x - 9) $

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#### 14. $ 4x + 18 $

- GCF = 2 → $ 2(2x + 9) $

✔ Answer: $ 2(2x + 9) $

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✔ Final Answers:

#### Part 1:
1. Done
2. Top: `x`, `4`; Dimensions: `5` and `(x+4)`; Product: `5(x+4)`; Sum: `5x+20`
3. Left: `6`; Dimensions: `6` and `(x+8)`; Product: `6(x+8)`; Sum: `6x+48`
4. Left: `10`, Top right: `3`; Dimensions: `10` and `(x+3)`; Product: `10(x+3)`; Sum: `10x+30`

#### Part 2:
5. $ 5(x + 7) $
6. $ 2(x + 6) $
7. $ 3(x - 7) $
8. $ 7(x - 3) $
9. $ -3(x + 5) $
10. $ -5(x - 9) $

#### Part 3:
11. $ 4(x - 4) $
12. $ -7(x + 5) $
13. $ 9(x - 9) $
14. $ 2(2x + 9) $

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