## Dividing Polynomials: (9x^3 - 18x^2 - x + 2) / (3x + 1)

This article will guide you through the process of dividing the polynomial (9x^3 - 18x^2 - x + 2) by the polynomial (3x + 1). We'll use the **long division method** for this.

### Understanding Long Division with Polynomials

Long division with polynomials works similarly to long division with numbers. Here's a breakdown of the process:

**Set up:**Write the dividend (9x^3 - 18x^2 - x + 2) inside the division symbol and the divisor (3x + 1) outside.**Divide:**Divide the leading term of the dividend (9x^3) by the leading term of the divisor (3x). This gives us 3x^2. Write this above the division symbol.**Multiply:**Multiply the divisor (3x + 1) by the term you just found (3x^2). This gives us 9x^3 + 3x^2. Write this below the dividend.**Subtract:**Subtract the result from the dividend. This will eliminate the leading term (9x^3).**Bring Down:**Bring down the next term from the dividend (-x).**Repeat:**Repeat steps 2-5 with the new polynomial.**Continue:**Keep repeating the process until you reach a remainder that has a degree less than the divisor.

### Let's Do It!

```
3x^2 - 7x + 2
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3x+1 | 9x^3 - 18x^2 - x + 2
-(9x^3 + 3x^2)
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-21x^2 - x
-(-21x^2 - 7x)
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6x + 2
-(6x + 2)
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0
```

### Explanation:

**Step 1:**We set up the division problem as shown above.**Step 2:**We divide 9x^3 by 3x, getting 3x^2.**Step 3:**Multiply (3x + 1) by 3x^2 to get 9x^3 + 3x^2.**Step 4:**Subtract (9x^3 + 3x^2) from the dividend.**Step 5:**Bring down -x.**Step 6:**Repeat the process. We divide -21x^2 by 3x, getting -7x. Multiply (3x + 1) by -7x and subtract from the polynomial.**Step 7:**Bring down 2. Repeat the process. We divide 6x by 3x, getting 2. Multiply (3x + 1) by 2 and subtract.

We end up with a remainder of 0.

### Conclusion

Therefore, (9x^3 - 18x^2 - x + 2) divided by (3x + 1) is **3x^2 - 7x + 2**. This means we can rewrite the original expression as:

**(9x^3 - 18x^2 - x + 2) = (3x + 1)(3x^2 - 7x + 2)**