Division of a polynomial by another polynomial | Part - 3

How to Divide a Polynomial by a Lower-Degree Polynomial When Terms Are Missing or Unordered

Dividing a polynomial by another of lower degree is a common task in algebra. However, the process can become confusing when the dividend (the polynomial being divided) has missing terms or the terms are not written in descending order of exponents. In this article, we’ll walk through a clear, step-by-step method to tackle such problems accurately using polynomial long division.

Understanding the Situation

Before you begin dividing, it’s essential to organize the polynomial properly. A standard polynomial is written in descending order of powers, such as:

  axⁿ + bxⁿ⁻¹ + ... + constant
  

But you may encounter expressions like:

(6x² + 3 - x⁴) ÷ (x + 1)

Here, the dividend is not in standard form, and some terms are missing. This can make long division difficult unless it's addressed first.

Step 1: Arrange the Polynomial in Standard Form

Reorder the terms of the dividend in descending powers of x and include any missing terms using zero coefficients:

  Dividend: -x⁴ + 0x³ + 6x² + 0x + 3
  Divisor: x + 1
  

Step 2: Use Polynomial Long Division

Now begin dividing just as you would with numbers, focusing on the leading terms:

1. Divide: -x⁴ ÷ x = -x³
2. Multiply: -x³ × (x + 1) = -x⁴ - x³
3. Subtract: (-x⁴ + 0x³) - (-x⁴ - x³) = x³
4. Bring down: 6x²
5. Divide: x³ ÷ x = x²
6. Multiply: x² × (x + 1) = x³ + x²
7. Subtract: (x³ + 6x²) - (x³ + x²) = 5x²
8. Bring down: 0x
9. Divide: 5x² ÷ x = 5x
10. Multiply: 5x × (x + 1) = 5x² + 5x
11. Subtract: (5x² + 0x) - (5x² + 5x) = -5x
12. Bring down: +3
13. Divide: -5x ÷ x = -5
14. Multiply: -5 × (x + 1) = -5x - 5
15. Subtract: (-5x + 3) - (-5x - 5) = 8

Quotient: -x³ + x² + 5x - 5

Remainder: 8

Final Answer: (-x⁴ + 6x² + 3) ÷ (x + 1) = -x³ + x² + 5x - 5 + 8/(x + 1)

Key Tips

• Arrange the polynomial terms from highest to lowest degree before beginning the division process.
• Insert zero placeholders for any missing terms. This keeps the terms properly aligned and reduces mistakes when subtracting.
• Keep your work organized by lining up terms based on their degrees.
• Check your answer by multiplying the quotient and divisor, then adding the remainder.

Practice Example

Try this one:

(2x³ + 7 - x) ÷ (x - 2)

First rewrite: 2x³ + 0x² - x + 7

Then apply the long division method using the steps above.

Conclusion

When dividing a polynomial by one of lower degree, it's crucial to prepare the dividend by arranging terms in descending order and filling in any missing powers with zero coefficients. Once this is done, standard polynomial long division can be applied confidently. With practice, even complex-looking expressions can be handled with ease.