Synthetic Division of Polynomials- Step-by-Step Process
What Synthetic Division Actually Is
Synthetic division is a shortcut method for dividing polynomials when you're working with a linear divisor. It works specifically when the divisor is in the form (x - c) or (x + c).
You could use long polynomial division every single time. But synthetic division is faster, requires less writing, and is less prone to arithmetic errors. If you've ever spent 10 minutes on polynomial division only to make a sign error somewhere, you already know why shortcuts exist.
When Synthetic Division Works (And When It Doesn't)
This method only applies in specific situations:
- The divisor must be linear — something like x - 3, x + 2, or x + 7
- The coefficient of x must be exactly 1
- You're dividing by something like (x - c), not (2x + 1)
If your divisor has a coefficient other than 1 on x, or isn't linear, go back to long division. There's no synthetic shortcut for everything.
The Setup: What Goes In Your Table
Before you start calculating, you need the right setup. This is where most people mess up.
Take your divisor (x - c) and find c. For (x - 3), c = 3. For (x + 5), c = -5. The sign flips.
Write c in the top right corner of your box. Then write all coefficients of your dividend polynomial in descending order of degree. Include zeros for any missing terms.
Example Setup
Dividing (2x³ + 3x² - 8x + 7) by (x - 2):
c = 2. Coefficients: 2, 3, -8, 7.
Your table looks like this before you start:
| 2 3 -8 7 | ||||
|---|---|---|---|---|
| 2 | ||||
Step-by-Step Synthetic Division Process
Here's the actual procedure. Five steps. That's it.
Step 1: Bring Down the First Coefficient
Take the leftmost coefficient and drop it straight down below the line. This becomes the first coefficient of your quotient.
Step 2: Multiply by c
Take that dropped number and multiply it by c. Write the result under the next column.
Step 3: Add Down
Add the numbers in that column. Write the sum below the line.
Step 4: Repeat
Multiply the new sum by c. Add it to the next column. Keep going until you've processed every column.
Step 5: Read Your Results
The numbers below the line are your answer. The last number is your remainder. The rest form the coefficients of your quotient polynomial.
Full Worked Example
Divide (2x³ + 3x² - 8x + 7) by (x - 2).
Setup: c = 2. Coefficients: 2, 3, -8, 7.
| 2 | 3 | -8 | 7 | |
|---|---|---|---|---|
| 2 | 2 | 7 | 6 | 19 |
Here's what happened at each step:
- Brought down 2
- 2 × 2 = 4. Added to 3. Got 7.
- 7 × 2 = 14. Added to -8. Got 6.
- 6 × 2 = 12. Added to 7. Got 19.
The bottom row: 2, 7, 6, 19
The quotient is 2x² + 7x + 6. The remainder is 19.
Verify by multiplying back: (x - 2)(2x² + 7x + 6) + 19 should equal the original polynomial.
Synthetic Division vs. Long Division
Here's the direct comparison:
| Feature | Synthetic Division | Long Division |
|---|---|---|
| Speed | Fast | Slow |
| Best for | Linear divisors (x - c) | Any divisor |
| Writing space needed | Small table | Full polynomial layout |
| Error rate | Lower | Higher |
| Works with non-unit coefficients | No | Yes |
Use synthetic when you can. Use long division when you must.
Common Mistakes That Destroy Your Answer
- Getting c wrong. For (x + 4), c = -4, not 4. The sign always flips.
- Forgetting zero coefficients. If your polynomial is x³ + 2x + 5, that's missing x². You must write 0 for that coefficient.
- Arithmetic errors. Synthetic reduces errors compared to long division, but doesn't eliminate them. Check your multiplication.
- Misreading the remainder. The last number is always the remainder. Don't accidentally include it in your quotient.
Practical How-To: Quick Reference
When you need to divide a polynomial by (x - c) quickly:
- Find c from your divisor
- List all coefficients of the dividend (including zeros)
- Bring down the first coefficient
- Multiply by c, add to next column
- Repeat until done
- Last number = remainder, rest = quotient coefficients
That's the entire method. Practice with three or four problems and it'll click.
Why This Matters
Synthetic division isn't just busywork. It's used to evaluate polynomials (that's the Remainder Theorem), find roots, and simplify expressions before other operations. If you're working with any polynomial function, this is a tool you need in your toolkit.
The method is fast enough that using long division every time is genuinely a waste of time. Learn synthetic, use it when it applies, and move on.