Dividing Polynomials Worksheets - Free Printable

It looks like you've uploaded a worksheet titled "Dividing Polynomials" that contains several polynomial division problems. However, since I can't view images directly, I'll need you to provide the actual problems from the image so I can solve and explain them.

But based on the visible text in your message, here are the problems listed:

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Divide by synthetic division:

1) $ x^3 + 5x^2 - 8x - 16 \div (x - 2) $

2) $ 2x^3 + 7x^2 - 12x - 35 \div (x + 5) $

3) $ 2x^3 + x^2 + 5x - 10 \div (x - 2) $

4) $ 3x^3 + 11x^2 - 35x + 24 \div (x - 3) $

5) $ 2x^3 - 8x^2 + 5x - 40 \div (x - 4) $

6) $ 5x^3 + 10x^2 + 19x + 16 \div (x + 2) $

7) $ 3x^3 + 12x^2 + 40x + 10 \div (x + 5) $

8) $ 3x^3 - 8x^2 - 19x + 12 \div (x - 4) $

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I’ll solve each of these using synthetic division, which is a shortcut method for dividing polynomials by linear divisors of the form $ (x - c) $. Let’s go step-by-step.

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🔹 Problem 1:

$ x^3 + 5x^2 - 8x - 16 \div (x - 2) $

Step 1: Use $ c = 2 $ (since divisor is $ x - 2 $)

Set up synthetic division:

```
2 | 1 5 -8 -16
| 2 14 12
-------------------
1 7 6 -4
```

Quotient: $ x^2 + 7x + 6 $
Remainder: $ -4 $

✔ So,
$$
\frac{x^3 + 5x^2 - 8x - 16}{x - 2} = x^2 + 7x + 6 - \frac{4}{x - 2}
$$

---

🔹 Problem 2:

$ 2x^3 + 7x^2 - 12x - 35 \div (x + 5) $

Note: $ x + 5 = x - (-5) $ → use $ c = -5 $

```
-5 | 2 7 -12 -35
| -10 15 -15
---------------------
2 -3 3 -50
```

Quotient: $ 2x^2 - 3x + 3 $
Remainder: $ -50 $

✔ So,
$$
\frac{2x^3 + 7x^2 - 12x - 35}{x + 5} = 2x^2 - 3x + 3 - \frac{50}{x + 5}
$$

---

🔹 Problem 3:

$ 2x^3 + x^2 + 5x - 10 \div (x - 2) $

Use $ c = 2 $

```
2 | 2 1 5 -10
| 4 10 30
------------------
2 5 15 20
```

Quotient: $ 2x^2 + 5x + 15 $
Remainder: $ 20 $

✔ So,
$$
\frac{2x^3 + x^2 + 5x - 10}{x - 2} = 2x^2 + 5x + 15 + \frac{20}{x - 2}
$$

---

🔹 Problem 4:

$ 3x^3 + 11x^2 - 35x + 24 \div (x - 3) $

Use $ c = 3 $

```
3 | 3 11 -35 24
| 9 60 45
-------------------
3 20 25 69
```

Quotient: $ 3x^2 + 20x + 25 $
Remainder: $ 69 $

✔ So,
$$
\frac{3x^3 + 11x^2 - 35x + 24}{x - 3} = 3x^2 + 20x + 25 + \frac{69}{x - 3}
$$

---

🔹 Problem 5:

$ 2x^3 - 8x^2 + 5x - 40 \div (x - 4) $

Use $ c = 4 $

```
4 | 2 -8 5 -40
| 8 0 20
-------------------
2 0 5 -20
```

Quotient: $ 2x^2 + 0x + 5 = 2x^2 + 5 $
Remainder: $ -20 $

✔ So,
$$
\frac{2x^3 - 8x^2 + 5x - 40}{x - 4} = 2x^2 + 5 - \frac{20}{x - 4}
$$

---

🔹 Problem 6:

$ 5x^3 + 10x^2 + 19x + 16 \div (x + 2) $

$ x + 2 = x - (-2) $ → $ c = -2 $

```
-2 | 5 10 19 16
| -10 0 -38
-------------------
5 0 19 -22
```

Quotient: $ 5x^2 + 0x + 19 = 5x^2 + 19 $
Remainder: $ -22 $

✔ So,
$$
\frac{5x^3 + 10x^2 + 19x + 16}{x + 2} = 5x^2 + 19 - \frac{22}{x + 2}
$$

---

🔹 Problem 7:

$ 3x^3 + 12x^2 + 40x + 10 \div (x + 5) $

Use $ c = -5 $

```
-5 | 3 12 40 10
| -15 15 -275
-------------------
3 -3 55 -265
```

Quotient: $ 3x^2 - 3x + 55 $
Remainder: $ -265 $

✔ So,
$$
\frac{3x^3 + 12x^2 + 40x + 10}{x + 5} = 3x^2 - 3x + 55 - \frac{265}{x + 5}
$$

---

🔹 Problem 8:

$ 3x^3 - 8x^2 - 19x + 12 \div (x - 4) $

Use $ c = 4 $

```
4 | 3 -8 -19 12
| 12 16 -12
--------------------
3 4 -3 0
```

Quotient: $ 3x^2 + 4x - 3 $
Remainder: $ 0 $

✔ So, this divides evenly:
$$
\frac{3x^3 - 8x^2 - 19x + 12}{x - 4} = 3x^2 + 4x - 3
$$

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✔ Final Answers Summary:

1) $ x^2 + 7x + 6 - \frac{4}{x - 2} $
2) $ 2x^2 - 3x + 3 - \frac{50}{x + 5} $
3) $ 2x^2 + 5x + 15 + \frac{20}{x - 2} $
4) $ 3x^2 + 20x + 25 + \frac{69}{x - 3} $
5) $ 2x^2 + 5 - \frac{20}{x - 4} $
6) $ 5x^2 + 19 - \frac{22}{x + 2} $
7) $ 3x^2 - 3x + 55 - \frac{265}{x + 5} $
8) $ 3x^2 + 4x - 3 $ (exact division)

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Let me know if you'd like explanations of any steps or want to verify with long division!