division of polynomials Math Worksheets, Math Practice for Kids. - Free Printable

Let's solve each polynomial division problem step by step. We'll use polynomial long division for the problems that require it and factoring or simplifying where possible.

---

1.

$$
\frac{20x^3 + 20x^2 - 68x - 56}{4x + 8}
$$

Step 1: Factor numerator and denominator if possible.

First, factor out common terms:

Numerator: $20x^3 + 20x^2 - 68x - 56$

Factor by grouping:
$$
= (20x^3 + 20x^2) + (-68x - 56)
= 20x^2(x + 1) - 4(17x + 14)
$$
Not helpful. Try factoring out GCF.

GCF of coefficients: 4 → factor out 4:
$$
= 4(5x^3 + 5x^2 - 17x - 14)
$$

Denominator: $4x + 8 = 4(x + 2)$

So:
$$
\frac{4(5x^3 + 5x^2 - 17x - 14)}{4(x + 2)} = \frac{5x^3 + 5x^2 - 17x - 14}{x + 2}
$$

Now divide $5x^3 + 5x^2 - 17x - 14$ by $x + 2$ using long division.

```
5x^2 - 5x - 7
-------------------
x + 2 | 5x^3 + 5x^2 - 17x - 14
-(5x^3 + 10x^2)
-----------------
-5x^2 - 17x
-(-5x^2 - 10x)
----------------
-7x - 14
-(-7x - 14)
----------
0
```

✔ So, result is:
$$
5x^2 - 5x - 7
$$

---

2.

$$
\frac{36x^3 - 54x^2 + 30x}{6x}
$$

Divide each term by $6x$:

$$
= \frac{36x^3}{6x} - \frac{54x^2}{6x} + \frac{30x}{6x}
= 6x^2 - 9x + 5
$$

✔ Answer: $6x^2 - 9x + 5$

---

3.

$$
\frac{42x^3 + 48x^2 + 42x}{6x}
$$

Divide each term by $6x$:

$$
= \frac{42x^3}{6x} + \frac{48x^2}{6x} + \frac{42x}{6x}
= 7x^2 + 8x + 7
$$

✔ Answer: $7x^2 + 8x + 7$

---

4.

$$
\frac{12x^2 - 40x - 63}{2x - 9}
$$

Use long division.

Divide $12x^2$ by $2x$: $6x$

Multiply: $6x(2x - 9) = 12x^2 - 54x$

Subtract:
$$
(12x^2 - 40x) - (12x^2 - 54x) = 14x
$$

Bring down $-63$: $14x - 63$

Divide $14x$ by $2x$: $7$

Multiply: $7(2x - 9) = 14x - 63$

Subtract: $0$

✔ Answer: $6x + 7$

---

5.

$$
\frac{20x^3 + 77x^2 + 27x - 72}{5x + 8}
$$

Long division:

Divide $20x^3$ by $5x$: $4x^2$

Multiply: $4x^2(5x + 8) = 20x^3 + 32x^2$

Subtract:
$$
(20x^3 + 77x^2) - (20x^3 + 32x^2) = 45x^2
$$

Bring down $+27x$: $45x^2 + 27x$

Divide $45x^2$ by $5x$: $9x$

Multiply: $9x(5x + 8) = 45x^2 + 72x$

Subtract:
$$
(45x^2 + 27x) - (45x^2 + 72x) = -45x
$$

Bring down $-72$: $-45x - 72$

Divide $-45x$ by $5x$: $-9$

Multiply: $-9(5x + 8) = -45x - 72$

Subtract: $0$

✔ Answer: $4x^2 + 9x - 9$

---

6.

$$
\frac{16x^2 - 62x - 45}{2x - 9}
$$

Long division:

Divide $16x^2$ by $2x$: $8x$

Multiply: $8x(2x - 9) = 16x^2 - 72x$

Subtract:
$$
(16x^2 - 62x) - (16x^2 - 72x) = 10x
$$

Bring down $-45$: $10x - 45$

Divide $10x$ by $2x$: $5$

Multiply: $5(2x - 9) = 10x - 45$

Subtract: $0$

✔ Answer: $8x + 5$

---

7.

$$
\frac{56x^3 + 82x^2 + 93x + 45}{7x + 5}
$$

Long division:

Divide $56x^3$ by $7x$: $8x^2$

Multiply: $8x^2(7x + 5) = 56x^3 + 40x^2$

Subtract:
$$
(56x^3 + 82x^2) - (56x^3 + 40x^2) = 42x^2
$$

Bring down $+93x$: $42x^2 + 93x$

Divide $42x^2$ by $7x$: $6x$

Multiply: $6x(7x + 5) = 42x^2 + 30x$

Subtract:
$$
(42x^2 + 93x) - (42x^2 + 30x) = 63x
$$

Bring down $+45$: $63x + 45$

Divide $63x$ by $7x$: $9$

Multiply: $9(7x + 5) = 63x + 45$

Subtract: $0$

✔ Answer: $8x^2 + 6x + 9$

---

8.

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

Divide each term by $x$:

$$
= \frac{8x^3}{x} + \frac{2x^2}{x} - \frac{4x}{x}
= 8x^2 + 2x - 4
$$

✔ Answer: $8x^2 + 2x - 4$

---

9.

$$
\frac{4x^3 - 37x^2 + 52x - 15}{4x - 5}
$$

Long division:

Divide $4x^3$ by $4x$: $x^2$

Multiply: $x^2(4x - 5) = 4x^3 - 5x^2$

Subtract:
$$
(4x^3 - 37x^2) - (4x^3 - 5x^2) = -32x^2
$$

Bring down $+52x$: $-32x^2 + 52x$

Divide $-32x^2$ by $4x$: $-8x$

Multiply: $-8x(4x - 5) = -32x^2 + 40x$

Subtract:
$$
(-32x^2 + 52x) - (-32x^2 + 40x) = 12x
$$

Bring down $-15$: $12x - 15$

Divide $12x$ by $4x$: $3$

Multiply: $3(4x - 5) = 12x - 15$

Subtract: $0$

✔ Answer: $x^2 - 8x + 3$

---

10.

$$
\frac{35x^3 - 46x^2 + 43x - 20}{7x - 5}
$$

Long division:

Divide $35x^3$ by $7x$: $5x^2$

Multiply: $5x^2(7x - 5) = 35x^3 - 25x^2$

Subtract:
$$
(35x^3 - 46x^2) - (35x^3 - 25x^2) = -21x^2
$$

Bring down $+43x$: $-21x^2 + 43x$

Divide $-21x^2$ by $7x$: $-3x$

Multiply: $-3x(7x - 5) = -21x^2 + 15x$

Subtract:
$$
(-21x^2 + 43x) - (-21x^2 + 15x) = 28x
$$

Bring down $-20$: $28x - 20$

Divide $28x$ by $7x$: $4$

Multiply: $4(7x - 5) = 28x - 20$

Subtract: $0$

✔ Answer: $5x^2 - 3x + 4$

---

## ✔ Final Answers:

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

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