Quick Answer: Two Methods
To divide a polynomial by a binomial, you can use:
- Polynomial Long Division - works for any divisor
- Synthetic Division - shortcut, only works when dividing by (x - a)
Key Formula: Dividend = Divisor × Quotient + Remainder. Written as: P(x) = D(x) · Q(x) + R
Method 1: Polynomial Long Division
Works just like numerical long division. Follow these steps:
- Divide: Divide the leading term of the dividend by the leading term of the divisor
- Multiply: Multiply the entire divisor by that result
- Subtract: Subtract the result from the current polynomial
- Bring down: Bring down the next term
- Repeat until no terms remain
Example: Divide (2x² + 7x + 3) by (x + 3)
- Step 1: 2x² / x = 2x. Write 2x in the quotient.
- Step 2: 2x × (x + 3) = 2x² + 6x
- Step 3: (2x² + 7x) - (2x² + 6x) = x. Bring down +3.
- Step 4: x / x = 1. Write +1 in the quotient.
- Step 5: 1 × (x + 3) = x + 3. Subtract: (x + 3) - (x + 3) = 0.
- Result: 2x + 1 (remainder 0)
Try It: Divide (3x² + 10x + 8) by (x + 2).
3x²/x = 3x. Multiply: 3x(x + 2) = 3x² + 6x. Subtract: 4x + 8.
4x/x = 4. Multiply: 4(x + 2) = 4x + 8. Subtract: 0.
Answer: 3x + 4
3x²/x = 3x. Multiply: 3x(x + 2) = 3x² + 6x. Subtract: 4x + 8.
4x/x = 4. Multiply: 4(x + 2) = 4x + 8. Subtract: 0.
Answer: 3x + 4
Method 2: Synthetic Division (Shortcut)
Use this method only when the divisor is of the form (x - a). It's faster than long division.
Example: Divide (x³ - 6x² + 7x + 6) by (x - 1)
- Write the root of the divisor: x - 1 = 0, so a = 1
- Write the coefficients: 1 | -6 | 7 | 6
- Bring down the first coefficient: 1
- Multiply by a (1) and add to next: -6 + 1 = -5
- Multiply by a (1) and add to next: 7 + (-5) = 2
- Multiply by a (1) and add to next: 6 + 2 = 8 (remainder)
Quotient: x² - 5x + 2, Remainder: 8
When to Use Each Method
| Feature | Long Division | Synthetic Division |
|---|---|---|
| Works with | Any divisor | Only (x - a) form |
| Speed | Slower | Much faster |
| Setup | Write full expressions | Only coefficients needed |
| Best for | (2x + 1), (x² + 3), etc. | (x - 5), (x + 2), etc. |
Common Mistakes
- Missing terms: If the polynomial skips a power, use a 0 placeholder. For x³ + 1, write coefficients as 1, 0, 0, 1.
- Sign errors in subtraction: When subtracting in long division, change ALL signs of the bottom row.
- Wrong root in synthetic division: For (x + 3), use a = -3 (not +3). For (x - 5), use a = 5.
Remember: If the remainder is 0, the divisor is a factor of the polynomial. If P(x) / (x - a) has remainder 0, then (x - a) is a factor and P(a) = 0.
Try our Polynomial Calculator or Remainder Theorem guide for related topics.