Computation With Polynomials: Division, 57% OFF - Free Printable

Problem: Division Worksheet

The task is to solve the given polynomial division problems using either long division or synthetic division. Below, I will solve each problem step by step.

---

Problem 1: $(r^2 + 6r + 15) \div (r + 5)$

#### Solution:
We will use synthetic division since the divisor is of the form $(r + 5)$, which can be rewritten as $(r - (-5))$.

1. Set up synthetic division:
- The root of the divisor $r + 5 = 0$ is $r = -5$.
- Write down the coefficients of the dividend $r^2 + 6r + 15$: $[1, 6, 15]$.
- Write $-5$ on the left.

```
-5 | 1 6 15
| -5 -5
----------------
1 1 10
```

2. Perform synthetic division:
- Bring down the first coefficient: $1$.
- Multiply $1 \cdot (-5) = -5$ and add to the next coefficient: $6 + (-5) = 1$.
- Multiply $1 \cdot (-5) = -5$ and add to the next coefficient: $15 + (-5) = 10$.

3. Interpret the result:
- The quotient is $r + 1$ (from the coefficients $1, 1$).
- The remainder is $10$.

Thus, the result is:
$$
\boxed{r + 1 + \frac{10}{r + 5}}
$$

---

Problem 2: $(r^2 + 10r + 13) \div (r + 7)$

#### Solution:
Again, we use synthetic division. The root of the divisor $r + 7 = 0$ is $r = -7$.

1. Set up synthetic division:
- Write down the coefficients of the dividend $r^2 + 10r + 13$: $[1, 10, 13]$.
- Write $-7$ on the left.

```
-7 | 1 10 13
| -7 -21
----------------
1 3 -8
```

2. Perform synthetic division:
- Bring down the first coefficient: $1$.
- Multiply $1 \cdot (-7) = -7$ and add to the next coefficient: $10 + (-7) = 3$.
- Multiply $3 \cdot (-7) = -21$ and add to the next coefficient: $13 + (-21) = -8$.

3. Interpret the result:
- The quotient is $r + 3$ (from the coefficients $1, 3$).
- The remainder is $-8$.

Thus, the result is:
$$
\boxed{r + 3 - \frac{8}{r + 7}}
$$

---

Problem 3: $(n^3 - 5n^2 - 33n - 37) \div (n - 9)$

#### Solution:
We use synthetic division. The root of the divisor $n - 9 = 0$ is $n = 9$.

1. Set up synthetic division:
- Write down the coefficients of the dividend $n^3 - 5n^2 - 33n - 37$: $[1, -5, -33, -37]$.
- Write $9$ on the left.

```
9 | 1 -5 -33 -37
| 9 36 27
---------------------
1 4 3 -10
```

2. Perform synthetic division:
- Bring down the first coefficient: $1$.
- Multiply $1 \cdot 9 = 9$ and add to the next coefficient: $-5 + 9 = 4$.
- Multiply $4 \cdot 9 = 36$ and add to the next coefficient: $-33 + 36 = 3$.
- Multiply $3 \cdot 9 = 27$ and add to the next coefficient: $-37 + 27 = -10$.

3. Interpret the result:
- The quotient is $n^2 + 4n + 3$ (from the coefficients $1, 4, 3$).
- The remainder is $-10$.

Thus, the result is:
$$
\boxed{n^2 + 4n + 3 - \frac{10}{n - 9}}
$$

---

Problem 4: $(x^3 + 6x^2 - 30x + 102) \div (x + 10)$

#### Solution:
We use synthetic division. The root of the divisor $x + 10 = 0$ is $x = -10$.

1. Set up synthetic division:
- Write down the coefficients of the dividend $x^3 + 6x^2 - 30x + 102$: $[1, 6, -30, 102]$.
- Write $-10$ on the left.

```
-10 | 1 6 -30 102
| -10 40 -100
---------------------
1 -4 10 2
```

2. Perform synthetic division:
- Bring down the first coefficient: $1$.
- Multiply $1 \cdot (-10) = -10$ and add to the next coefficient: $6 + (-10) = -4$.
- Multiply $-4 \cdot (-10) = 40$ and add to the next coefficient: $-30 + 40 = 10$.
- Multiply $10 \cdot (-10) = -100$ and add to the next coefficient: $102 + (-100) = 2$.

3. Interpret the result:
- The quotient is $x^2 - 4x + 10$ (from the coefficients $1, -4, 10$).
- The remainder is $2$.

Thus, the result is:
$$
\boxed{x^2 - 4x + 10 + \frac{2}{x + 10}}
$$

---

Problem 5: $(2v^3 - 20v^2 + 56v - 46) \div (v - 6)$

#### Solution:
We use synthetic division. The root of the divisor $v - 6 = 0$ is $v = 6$.

1. Set up synthetic division:
- Write down the coefficients of the dividend $2v^3 - 20v^2 + 56v - 46$: $[2, -20, 56, -46]$.
- Write $6$ on the left.

```
6 | 2 -20 56 -46
| 12 -48 48
---------------------
2 -8 8 2
```

2. Perform synthetic division:
- Bring down the first coefficient: $2$.
- Multiply $2 \cdot 6 = 12$ and add to the next coefficient: $-20 + 12 = -8$.
- Multiply $-8 \cdot 6 = -48$ and add to the next coefficient: $56 + (-48) = 8$.
- Multiply $8 \cdot 6 = 48$ and add to the next coefficient: $-46 + 48 = 2$.

3. Interpret the result:
- The quotient is $2v^2 - 8v + 8$ (from the coefficients $2, -8, 8$).
- The remainder is $2$.

Thus, the result is:
$$
\boxed{2v^2 - 8v + 8 + \frac{2}{v - 6}}
$$

---

Problem 6: $(8r^3 - 49r^2 - 45r - 36) \div (r - 7)$

#### Solution:
We use synthetic division. The root of the divisor $r - 7 = 0$ is $r = 7$.

1. Set up synthetic division:
- Write down the coefficients of the dividend $8r^3 - 49r^2 - 45r - 36$: $[8, -49, -45, -36]$.
- Write $7$ on the left.

```
7 | 8 -49 -45 -36
| 56 49 28
---------------------
8 7 4 -8
```

2. Perform synthetic division:
- Bring down the first coefficient: $8$.
- Multiply $8 \cdot 7 = 56$ and add to the next coefficient: $-49 + 56 = 7$.
- Multiply $7 \cdot 7 = 49$ and add to the next coefficient: $-45 + 49 = 4$.
- Multiply $4 \cdot 7 = 28$ and add to the next coefficient: $-36 + 28 = -8$.

3. Interpret the result:
- The quotient is $8r^2 + 7r + 4$ (from the coefficients $8, 7, 4$).
- The remainder is $-8$.

Thus, the result is:
$$
\boxed{8r^2 + 7r + 4 - \frac{8}{r - 7}}
$$

---

Problem 7: $(m^3 - 20) \div (m - 3)$

#### Solution:
We use synthetic division. The root of the divisor $m - 3 = 0$ is $m = 3$.

1. Set up synthetic division:
- Write down the coefficients of the dividend $m^3 + 0m^2 + 0m - 20$: $[1, 0, 0, -20]$.
- Write $3$ on the left.

```
3 | 1 0 0 -20
| 3 9 27
---------------------
1 3 9 7
```

2. Perform synthetic division:
- Bring down the first coefficient: $1$.
- Multiply $1 \cdot 3 = 3$ and add to the next coefficient: $0 + 3 = 3$.
- Multiply $3 \cdot 3 = 9$ and add to the next coefficient: $0 + 9 = 9$.
- Multiply $9 \cdot 3 = 27$ and add to the next coefficient: $-20 + 27 = 7$.

3. Interpret the result:
- The quotient is $m^2 + 3m + 9$ (from the coefficients $1, 3, 9$).
- The remainder is $7$.

Thus, the result is:
$$
\boxed{m^2 + 3m + 9 + \frac{7}{m - 3}}
$$

---

Problem 8: $(2k^3 - 13k^2 - 77k + 60) \div (k - 10)$

#### Solution:
We use synthetic division. The root of the divisor $k - 10 = 0$ is $k = 10$.

1. Set up synthetic division:
- Write down the coefficients of the dividend $2k^3 - 13k^2 - 77k + 60$: $[2, -13, -77, 60]$.
- Write $10$ on the left.

```
10 | 2 -13 -77 60
| 20 70 23
---------------------
2 7 -7 83
```

2. Perform synthetic division:
- Bring down the first coefficient: $2$.
- Multiply $2 \cdot 10 = 20$ and add to the next coefficient: $-13 + 20 = 7$.
- Multiply $7 \cdot 10 = 70$ and add to the next coefficient: $-77 + 70 = -7$.
- Multiply $-7 \cdot 10 = -70$ and add to the next coefficient: $60 + (-70) = -10$.

3. Interpret the result:
- The quotient is $2k^2 + 7k - 7$ (from the coefficients $2, 7, -7$).
- The remainder is $-10$.

Thus, the result is:
$$
\boxed{2k^2 + 7k - 7 - \frac{10}{k - 10}}
$$

---

Final Answers:

1. $\boxed{r + 1 + \frac{10}{r + 5}}$
2. $\boxed{r + 3 - \frac{8}{r + 7}}$
3. $\boxed{n^2 + 4n + 3 - \frac{10}{n - 9}}$
4. $\boxed{x^2 - 4x + 10 + \frac{2}{x + 10}}$
5. $\boxed{2v^2 - 8v + 8 + \frac{2}{v - 6}}$
6. $\boxed{8r^2 + 7r + 4 - \frac{8}{r - 7}}$
7. $\boxed{m^2 + 3m + 9 + \frac{7}{m - 3}}$
8. $\boxed{2k^2 + 7k - 7 - \frac{10}{k - 10}}$