Math 11 Long and Synthetic Division Worksheet with eight polynomial division exercises.

Let's solve each of the polynomial division problems on the worksheet using long division or synthetic division, depending on what’s most efficient. I'll go through each problem step by step.

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

We use long division.

```
r + 1
_________
r + 5 | r² + 6r + 15
-(r² + 5r)
-----------
r + 15
-(r + 5)
--------
10
```

- Quotient: $r + 1$
- Remainder: $10$

✔ So,
$$
\frac{r^2 + 6r + 15}{r + 5} = r + 1 + \frac{10}{r + 5}
$$

---

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

Long division:

```
r + 3
_________
r + 7 | r² + 10r + 13
-(r² + 7r)
-----------
3r + 13
-(3r + 21)
--------
-8
```

- Quotient: $r + 3$
- Remainder: $-8$

✔ So,
$$
\frac{r^2 + 10r + 13}{r + 7} = r + 3 - \frac{8}{r + 7}
$$

---

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

Use synthetic division since divisor is linear: $n - 9$ → use root $9$

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

- Quotient: $n^2 + 4n + 3$
- Remainder: $-10$

✔ So,
$$
\frac{n^3 - 5n^2 - 33n - 37}{n - 9} = n^2 + 4n + 3 - \frac{10}{n - 9}
$$

---

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

Divisor: $x + 10$, so root = $-10$

Synthetic division:

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

- Quotient: $x^2 - 4x + 10$
- Remainder: $2$

✔ So,
$$
\frac{x^3 + 6x^2 - 30x + 102}{x + 10} = x^2 - 4x + 10 + \frac{2}{x + 10}
$$

---

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

Root: $6$

Synthetic division:

```
| 2 -20 56 -46
6 | 12 -24 72
-----------------------
| 2 -8 32 26
```

- Quotient: $2v^2 - 8v + 32$
- Remainder: $26$

✔ So,
$$
\frac{2v^3 - 20v^2 + 56v - 46}{v - 6} = 2v^2 - 8v + 32 + \frac{26}{v - 6}
$$

---

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

Root: $7$

Synthetic division:

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

- Quotient: $8r^2 + 7r + 4$
- Remainder: $-8$

✔ So,
$$
\frac{8r^3 - 49r^2 - 45r - 36}{r - 7} = 8r^2 + 7r + 4 - \frac{8}{r - 7}
$$

---

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

Note: This is a cubic with missing terms → write as: $m^3 + 0m^2 + 0m - 20$

Root: $3$

Synthetic division:

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

- Quotient: $m^2 + 3m + 9$
- Remainder: $7$

✔ So,
$$
\frac{m^3 - 20}{m - 3} = m^2 + 3m + 9 + \frac{7}{m - 3}
$$

---

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

Root: $10$

Synthetic division:

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

- Quotient: $2k^2 + 7k - 7$
- Remainder: $-10$

✔ So,
$$
\frac{2k^3 - 13k^2 - 77k + 60}{k - 10} = 2k^2 + 7k - 7 - \frac{10}{k - 10}
$$

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✔ Final Answers Summary:

1. $r + 1 + \frac{10}{r + 5}$
2. $r + 3 - \frac{8}{r + 7}$
3. $n^2 + 4n + 3 - \frac{10}{n - 9}$
4. $x^2 - 4x + 10 + \frac{2}{x + 10}$
5. $2v^2 - 8v + 32 + \frac{26}{v - 6}$
6. $8r^2 + 7r + 4 - \frac{8}{r - 7}$
7. $m^2 + 3m + 9 + \frac{7}{m - 3}$
8. $2k^2 + 7k - 7 - \frac{10}{k - 10}$

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