Solving Problems Involving Polynomial Division and Factor Theorem ... - Free Printable

We are given:

- The volume of a swimming pool:
$$
V = 3x^3 - 4x^2 - 28x - 16
$$
- The height of the pool:
$$
h = x - 4
$$

We are to find the dimensions of the rectangular base, which means we need to find the length and width such that:
$$
V = \text{length} \times \text{width} \times \text{height}
$$
So,
$$
\text{Base Area} = \frac{V}{h} = \frac{3x^3 - 4x^2 - 28x - 16}{x - 4}
$$

We'll use polynomial division or the Factor Theorem to divide the volume polynomial by $x - 4$.

---

Step 1: Use the Factor Theorem (Synthetic Division)

We divide $3x^3 - 4x^2 - 28x - 16$ by $x - 4$ using synthetic division.

Set up synthetic division with root $x = 4$:

```
4 | 3 -4 -28 -16
| 12 32 16
---------------------
3 8 4 0
```

Explanation:
- Bring down 3.
- Multiply: $3 \times 4 = 12$, add to next coefficient: $-4 + 12 = 8$
- Multiply: $8 \times 4 = 32$, add to next: $-28 + 32 = 4$
- Multiply: $4 \times 4 = 16$, add to last: $-16 + 16 = 0$

Remainder is 0, so $x - 4$ is a factor.

The quotient is:
$$
3x^2 + 8x + 4
$$

So,
$$
\frac{3x^3 - 4x^2 - 28x - 16}{x - 4} = 3x^2 + 8x + 4
$$

Now, we need to factor $3x^2 + 8x + 4$ to get the dimensions of the base.

---

Step 2: Factor $3x^2 + 8x + 4$

We look for two numbers that multiply to $3 \times 4 = 12$ and add to 8.

Factors of 12:
- 1 & 12 → sum 13
- 2 & 6 → sum 8 ✔
- 3 & 4 → sum 7

So, split the middle term:
$$
3x^2 + 6x + 2x + 4
$$
Group:
$$
(3x^2 + 6x) + (2x + 4) = 3x(x + 2) + 2(x + 2)
$$
Factor:
$$
(3x + 2)(x + 2)
$$

So,
$$
3x^2 + 8x + 4 = (x + 2)(3x + 2)
$$

---

Final Answer:

The base area is $(x + 2)(3x + 2)$, and height is $x - 4$.

So, the dimensions of the rectangular base are:
$$
(x + 2) \text{ and } (3x + 2)
$$

That matches Option (A).

---

✔ Correct Answer: (A) (x + 2)(3x + 2)

---

Summary:

- Volume = $3x^3 - 4x^2 - 28x - 16$
- Height = $x - 4$
- Divide volume by height → get $3x^2 + 8x + 4$
- Factor it → $(x + 2)(3x + 2)$
- So base dimensions are $(x + 2)$ and $(3x + 2)$

✔ Answer: (A)