This algebra worksheet offers ten practice problems for dividing polynomials by both monomials and binomials.

Let's solve each polynomial division problem step by step. We'll use polynomial long division or factoring/simplifying where applicable.

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1. $\frac{45x^3 + 72x^2 + 9x}{9x}$

We can factor out $9x$ from the numerator:

$$
= \frac{9x(5x^2 + 8x + 1)}{9x} = 5x^2 + 8x + 1
$$

✔ Answer: $5x^2 + 8x + 1$

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2. $\frac{40x^2 + 41x - 21}{8x - 3}$

Use polynomial long division:

Divide $40x^2$ by $8x$:
→ $5x$

Multiply: $5x(8x - 3) = 40x^2 - 15x$

Subtract:
$(40x^2 + 41x - 21) - (40x^2 - 15x) = 56x - 21$

Now divide $56x$ by $8x$: → $7$

Multiply: $7(8x - 3) = 56x - 21$

Subtract: $(56x - 21) - (56x - 21) = 0$

✔ Answer: $5x + 7$

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3. $\frac{12x^3 + 8x^2 - 40x - 16}{2x + 4}$

First, factor denominator: $2x + 4 = 2(x + 2)$

Try factoring numerator:

Factor out 4:
$12x^3 + 8x^2 - 40x - 16 = 4(3x^3 + 2x^2 - 10x - 4)$

Try to factor $3x^3 + 2x^2 - 10x - 4$. Try rational root theorem: possible roots ±1, ±2, ±4, ±1/3, etc.

Try $x = -2$:
$3(-8) + 2(4) -10(-2) -4 = -24 + 8 + 20 - 4 = 0$ → Yes!

So $x + 2$ is a factor.

Use synthetic division on $3x^3 + 2x^2 -10x -4$ with root $-2$:

```
-2 | 3 2 -10 -4
| -6 8 4
-------------------
3 -4 -2 0
```

So quotient is $3x^2 - 4x - 2$

Thus:
$$
\frac{12x^3 + 8x^2 - 40x - 16}{2x + 4} = \frac{4(x+2)(3x^2 - 4x - 2)}{2(x+2)} = \frac{4}{2}(3x^2 - 4x - 2) = 2(3x^2 - 4x - 2)
$$

✔ Answer: $6x^2 - 8x - 4$

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4. $\frac{8x^3 + 14x^2 - 25x - 25}{2x + 5}$

Use long division:

Divide $8x^3$ by $2x$ → $4x^2$

Multiply: $4x^2(2x + 5) = 8x^3 + 20x^2$

Subtract:
$(8x^3 + 14x^2) - (8x^3 + 20x^2) = -6x^2$

Bring down $-25x$: → $-6x^2 - 25x$

Divide $-6x^2$ by $2x$ → $-3x$

Multiply: $-3x(2x + 5) = -6x^2 - 15x$

Subtract: $(-6x^2 - 25x) - (-6x^2 - 15x) = -10x$

Bring down $-25$: → $-10x - 25$

Divide $-10x$ by $2x$ → $-5$

Multiply: $-5(2x + 5) = -10x - 25$

Subtract: $(-10x - 25) - (-10x - 25) = 0$

✔ Answer: $4x^2 - 3x - 5$

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5. $\frac{6x^3 + 13x^2 - 10x + 15}{x + 3}$

Use synthetic division with root $-3$:

```
-3 | 6 13 -10 15
| -18 15 -15
---------------------
6 -5 5 0
```

Quotient: $6x^2 - 5x + 5$, remainder 0

✔ Answer: $6x^2 - 5x + 5$

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6. $\frac{18x^3 + 21x^2 + 27x}{3x}$

Factor numerator: $3x(6x^2 + 7x + 9)$

$$
= \frac{3x(6x^2 + 7x + 9)}{3x} = 6x^2 + 7x + 9
$$

✔ Answer: $6x^2 + 7x + 9$

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7. $\frac{12x^3 - 6x^2 + 16x}{2x}$

Factor numerator: $2x(6x^2 - 3x + 8)$

$$
= \frac{2x(6x^2 - 3x + 8)}{2x} = 6x^2 - 3x + 8
$$

✔ Answer: $6x^2 - 3x + 8$

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8. $\frac{4x^3 - 16x^2 + 12x}{2x}$

Factor numerator: $2x(2x^2 - 8x + 6)$

$$
= \frac{2x(2x^2 - 8x + 6)}{2x} = 2x^2 - 8x + 6
$$

✔ Answer: $2x^2 - 8x + 6$

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9. $\frac{12x^3 + 30x^2 - 30x}{6x}$

Factor numerator: $6x(2x^2 + 5x - 5)$

$$
= \frac{6x(2x^2 + 5x - 5)}{6x} = 2x^2 + 5x - 5
$$

✔ Answer: $2x^2 + 5x - 5$

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10. $\frac{54x^2 + 33x + 3}{9x + 1}$

Use long division:

Divide $54x^2$ by $9x$ → $6x$

Multiply: $6x(9x + 1) = 54x^2 + 6x$

Subtract: $(54x^2 + 33x) - (54x^2 + 6x) = 27x$

Bring down $+3$: → $27x + 3$

Divide $27x$ by $9x$ → $3$

Multiply: $3(9x + 1) = 27x + 3$

Subtract: $0$

✔ Answer: $6x + 3$

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## ✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $5x^2 + 8x + 1$ |
| 2 | $5x + 7$ |
| 3 | $6x^2 - 8x - 4$ |
| 4 | $4x^2 - 3x - 5$ |
| 5 | $6x^2 - 5x + 5$ |
| 6 | $6x^2 + 7x + 9$ |
| 7 | $6x^2 - 3x + 8$ |
| 8 | $2x^2 - 8x + 6$ |
| 9 | $2x^2 + 5x - 5$ |
| 10 | $6x + 3$ |

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