Polynomial division practice worksheet with eight problems, designed for educational use.

Let's solve each polynomial division problem step by step using long division or synthetic division where appropriate. We'll go through each one carefully.

---

1) Divide: $ \frac{x^2 - 13x + 40}{x - 8} $

We use polynomial long division:

```
x - 5
_________
x - 8 | x² -13x +40
-(x² - 8x)
-----------
-5x +40
-(-5x +40)
-----------
0
```

✔ Quotient: $ x - 5 $, Remainder: 0
So: $ \frac{x^2 - 13x + 40}{x - 8} = x - 5 $

---

2) Divide: $ \frac{12x^2 - 39x + 30}{4x - 5} $

Long division:

```
3x - 6
_________
4x - 5 | 12x² -39x +30
-(12x² -15x)
-------------
-24x +30
-(-24x +30)
-------------
0
```

✔ Quotient: $ 3x - 6 $, Remainder: 0
So: $ \frac{12x^2 - 39x + 30}{4x - 5} = 3x - 6 $

---

3) Divide: $ \frac{14x^2 + 69x + 27}{7x + 3} $

```
2x + 9
_________
7x + 3 | 14x² +69x +27
-(14x² +6x)
------------
63x +27
-(63x +27)
------------
0
```

✔ Quotient: $ 2x + 9 $, Remainder: 0
So: $ \frac{14x^2 + 69x + 27}{7x + 3} = 2x + 9 $

---

4) Divide: $ \frac{6x^3 + 23x^2 + 10x + 1}{3x + 1} $

Use long division:

```
2x² + 7x + 1
_______________
3x + 1 | 6x³ +23x² +10x +1
-(6x³ + 2x²)
--------------
21x² +10x
-(21x² +7x)
------------
3x +1
-(3x +1)
-------
0
```

✔ Quotient: $ 2x^2 + 7x + 1 $, Remainder: 0
So: $ \frac{6x^3 + 23x^2 + 10x + 1}{3x + 1} = 2x^2 + 7x + 1 $

---

5) Divide: $ \frac{3x^2 - 5x - 22}{x - 2} $

Long division:

```
3x + 1
_________
x - 2 | 3x² -5x -22
-(3x² -6x)
-----------
x -22
-(x -2)
--------
-20
```

✔ Quotient: $ 3x + 1 $, Remainder: $ -20 $
So: $ \frac{3x^2 - 5x - 22}{x - 2} = 3x + 1 - \frac{20}{x - 2} $

---

6) Divide: $ \frac{7x^2 + 25x + 12}{x - 3} $

```
7x + 46
_________
x - 3 | 7x² +25x +12
-(7x² -21x)
------------
46x +12
-(46x -138)
------------
150
```

Wait! Let’s double-check that subtraction:

After $ 46x + 12 $, subtract $ (46x - 138) $ → $ 46x + 12 - 46x + 138 = 150 $

✔ Quotient: $ 7x + 46 $, Remainder: $ 150 $

So: $ \frac{7x^2 + 25x + 12}{x - 3} = 7x + 46 + \frac{150}{x - 3} $

But wait — let's verify if this is correct by plugging in $ x = 3 $ into the original expression:

Numerator at $ x = 3 $: $ 7(9) + 25(3) + 12 = 63 + 75 + 12 = 150 $

Denominator: $ 3 - 3 = 0 $ → undefined, but remainder should be $ f(3) = 150 $, which matches.

So yes, correct.

✔ Final: $ 7x + 46 + \frac{150}{x - 3} $

---

7) Divide: $ \frac{5x^3 + 4x^2 - 45x - 36}{x^2 - 9} $

Note: $ x^2 - 9 = (x - 3)(x + 3) $

We divide a cubic by a quadratic.

Use long division:

```
5x + 4
_________
x² - 9 | 5x³ +4x² -45x -36
-(5x³ -45x)
------------
4x² +0x -36
-(4x² -36)
------------
0
```

Wait — let’s do it properly:

Divide $ 5x^3 $ by $ x^2 $ → $ 5x $

Multiply: $ 5x(x^2 - 9) = 5x^3 - 45x $

Subtract:

$$
(5x^3 + 4x^2 - 45x - 36) - (5x^3 - 45x) = 4x^2 + 0x - 36
$$

Now divide $ 4x^2 $ by $ x^2 $ → $ 4 $

Multiply: $ 4(x^2 - 9) = 4x^2 - 36 $

Subtract:

$$
(4x^2 + 0x - 36) - (4x^2 - 36) = 0
$$

✔ Quotient: $ 5x + 4 $, Remainder: 0
So: $ \frac{5x^3 + 4x^2 - 45x - 36}{x^2 - 9} = 5x + 4 $

---

8) Divide: $ \frac{5x^5 - 45x^4 + 52x^3 - 18x^2 + 20}{5x^2 + 2} $

This is more complex. Use long division.

We divide a degree-5 polynomial by a degree-2 polynomial.

Step-by-step:

#### Step 1: $ \frac{5x^5}{5x^2} = x^3 $

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

Subtract:

$$
(5x^5 - 45x^4 + 52x^3 - 18x^2 + 20) - (5x^5 + 2x^3) = -45x^4 + 50x^3 - 18x^2 + 20
$$

#### Step 2: $ \frac{-45x^4}{5x^2} = -9x^2 $

Multiply: $ -9x^2(5x^2 + 2) = -45x^4 - 18x^2 $

Subtract:

$$
(-45x^4 + 50x^3 - 18x^2 + 20) - (-45x^4 - 18x^2) = 50x^3 + 0x^2 + 20
$$

#### Step 3: $ \frac{50x^3}{5x^2} = 10x $

Multiply: $ 10x(5x^2 + 2) = 50x^3 + 20x $

Subtract:

$$
(50x^3 + 0x^2 + 20) - (50x^3 + 20x) = 0x^3 + 0x^2 - 20x + 20
$$

#### Step 4: $ \frac{-20x}{5x^2} = -\frac{4}{x} $ → not valid for polynomial division.

So we stop here.

Current quotient: $ x^3 - 9x^2 + 10x $

Remainder: $ -20x + 20 $

So:

$$
\frac{5x^5 - 45x^4 + 52x^3 - 18x^2 + 20}{5x^2 + 2} = x^3 - 9x^2 + 10x + \frac{-20x + 20}{5x^2 + 2}
$$

We can factor numerator of remainder: $ -20x + 20 = -20(x - 1) $

So final answer:

✔ $ x^3 - 9x^2 + 10x - \frac{20(x - 1)}{5x^2 + 2} $

---

✔ Final Answers Summary:

1) $ x - 5 $
2) $ 3x - 6 $
3) $ 2x + 9 $
4) $ 2x^2 + 7x + 1 $
5) $ 3x + 1 - \frac{20}{x - 2} $
6) $ 7x + 46 + \frac{150}{x - 3} $
7) $ 5x + 4 $
8) $ x^3 - 9x^2 + 10x - \frac{20(x - 1)}{5x^2 + 2} $

Let me know if you'd like these checked with synthetic division (where applicable) or graphed!