Dividing Polynomials: Step-by-step long division example on a whiteboard.

You're asked to divide the polynomial 2x² + 7x – 4 by x – 3 using long division. The steps provided — Divide, Multiply, Subtract, Bring Down — are the core of polynomial long division.

Let’s solve it step-by-step:

---

Step 1: Set up the division

We’re dividing:
```
_______________
x - 3 ) 2x² + 7x - 4
```

---

Step 2: Divide the leading terms

Divide the first term of the dividend (2x²) by the first term of the divisor (x):

> 2x² ÷ x = 2x

Write 2x above the division bar, aligned with the x term.

```
2x
_______________
x - 3 ) 2x² + 7x - 4
```

---

Step 3: Multiply the divisor by this result

Multiply 2x by (x – 3):

> 2x · (x – 3) = 2x² – 6x

Write this under the dividend:

```
2x
_______________
x - 3 ) 2x² + 7x - 4
2x² - 6x
```

---

Step 4: Subtract

Subtract (2x² – 6x) from (2x² + 7x):

> (2x² + 7x) – (2x² – 6x) = 0x² + 13x

Bring down the next term (–4):

```
2x
_______________
x - 3 ) 2x² + 7x - 4
2x² - 6x
----------
13x - 4
```

---

Step 5: Repeat the process — Divide again

Divide the new leading term (13x) by x:

> 13x ÷ x = 13

Write +13 above the division bar, next to 2x:

```
2x + 13
_______________
x - 3 ) 2x² + 7x - 4
2x² - 6x
----------
13x - 4
```

---

Step 6: Multiply the divisor by 13

Multiply 13 · (x – 3) = 13x – 39

Write it under 13x – 4:

```
2x + 13
_______________
x - 3 ) 2x² + 7x - 4
2x² - 6x
----------
13x - 4
13x - 39
```

---

Step 7: Subtract

Subtract (13x – 39) from (13x – 4):

> (13x – 4) – (13x – 39) = 0x + 35

So the remainder is 35.

```
2x + 13
_______________
x - 3 ) 2x² + 7x - 4
2x² - 6x
----------
13x - 4
13x - 39
----------
35
```

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✔ Final Answer:

> Quotient: 2x + 13, Remainder: 35

We write the final answer as:

> 2x + 13 + 35/(x – 3)

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💡 Check (Optional)

To verify, multiply the quotient by the divisor and add the remainder:

> (2x + 13)(x – 3) + 35
= 2x(x – 3) + 13(x – 3) + 35
= 2x² – 6x + 13x – 39 + 35
= 2x² + 7x – 4 ✔

Matches the original dividend!

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📌 Summary:

- Divide: 2x² ÷ x → 2x
- Multiply: 2x(x – 3) → 2x² – 6x
- Subtract: (2x² + 7x) – (2x² – 6x) → 13x
- Bring Down: Bring down –4 → 13x – 4
- Repeat: 13x ÷ x → 13; 13(x – 3) → 13x – 39; subtract → 35 (remainder)

✔ Answer: 2x + 13 + 35/(x – 3)