Division of a polynomial by another polynomial | Part - 2

How to Divide a Polynomial by Another Polynomial When Terms Are Out of Order or Missing

Polynomial division is a key concept in algebraic operations. However, the process can be confusing when the powers of the variable in the dividend are either disordered or some terms are missing. This article will guide you through the correct steps to divide such polynomials using long division, ensuring accuracy even when the expression isn’t neatly arranged.

Understanding the Problem

Consider the polynomial division:

(4x³ + 5 - x²) ÷ (x - 2)

At first glance, the dividend (the expression being divided) is not in standard form because the powers of x are out of order: the x² term comes after the constant, and it's also missing the x¹ term.

Step 1: Rearrange and Fill in the Missing Terms

Before dividing, rewrite the polynomial in descending order of powers, inserting any missing terms with a coefficient of zero:

  Dividend: 4x³ - x² + 0x + 5
  Divisor: x - 2
  

This setup ensures you can align like terms during division.

Step 2: Perform Polynomial Long Division

Now use polynomial long division as you would with numbers. Start by dividing the first term of the dividend by the first term of the divisor:

1. Divide: 4x³ ÷ x = 4x²
2. Multiply: 4x² × (x - 2) = 4x³ - 8x²
3. Subtract: (4x³ - x²) - (4x³ - 8x²) = 7x²
4. Bring down: 0x
5. Repeat: 7x² ÷ x = 7x
6. Multiply: 7x × (x - 2) = 7x² - 14x
7. Subtract: (7x² + 0x) - (7x² - 14x) = 14x
8. Bring down: +5
9. Divide: 14x ÷ x = 14
10. Multiply: 14 × (x - 2) = 14x - 28
11. Subtract: (14x + 5) - (14x - 28) = 33

Quotient: 4x² + 7x + 14

Remainder: 33

So, the result of the division is:

(4x³ + 5 - x²) ÷ (x - 2) = 4x² + 7x + 14 + 33/(x - 2)

Tips for Success

• Always write the dividend in descending order of the variable's exponents.
• Insert missing terms with a coefficient of 0 to maintain structure during subtraction.
• Align like terms vertically to avoid confusion.
• Double-check signs during subtraction—it’s the most common place for mistakes.

Dividing polynomials becomes manageable and systematic when you arrange terms correctly and fill in the gaps with zeroes. Following these steps ensures you stay organized even when terms are missing or out of order. With consistent practice, you’ll gain confidence in handling even the messiest-looking polynomial divisions.