Synthetic Division vs Polynomial Long Division
Same problem both ways: see why synthetic division is faster, when you must use polynomial long division instead, and how the two layouts line up.
Synthetic Division vs Polynomial Long Division
When you divide a polynomial by a linear binomial of the form x - c, synthetic division and polynomial long division produce the same quotient and remainder.That speed advantage grows as the polynomial gets longer. But synthetic division stops working the moment the divisor is quadratic or higher, then only long division applies. The decision between them is a single check: is the divisor x - c? If yes, use synthetic division. If no, if the divisor is something like x² + 1, use long division. That is the rule.
One Problem Solved Both Ways Side by Side
Divide 3x³ + 5x² − 2x + 7 by x − 2. Here both methods work, and the results match exactly.
Long Division Walkthrough
Set up the division: 3x³ + 5x² − 2x + 7 divided by x − 2. The leading term 3x³ divided by x gives 3x². Multiply the divisor by 3x²: 3x³ − 6x². Subtract from the dividend: 5x² − (−6x²) = 11x². Bring down the next term. Repeat: 11x² divided by x gives 11x. Multiply 11x(x − 2) = 11x² − 22x. Subtract: −2x − (−22x) = 20x. Bring down the constant. 20x divided by x gives 20. Multiply 20(x − 2) = 20x − 40. Subtract: 7 − (−40) = 47. The quotient is 3x² + 11x + 20, remainder 47. Written as 3x² + 11x + 20 + 47/(x − 2).
Synthetic Division Procedure
Write the coefficients of the dividend: 3, 5, −2, 7. For divisor x − 2, c = 2. Bring down the leading coefficient 3. Multiply 3 by 2: 6. Add to the next coefficient: 5 + 6 = 11. Multiply 11 by 2: 22. Add to the next: −2 + 22 = 20. Multiply 20 by 2: 40. Add to the last: 7 + 40 = 47. The bottom row reads 3, 11, 20, 47. The first three numbers are the coefficients of the quotient: 3x² + 11x + 20. The last number is the remainder: 47. Identical result, fewer written steps.
How the Synthetic Table Maps onto Long Division
The synthetic division table is a compressed version of long division. Each bring-down, multiply, and add step in the synthetic table corresponds to one cycle of dividing, multiplying, and subtracting in long division. The leading coefficient in the synthetic table is the same as the first quotient term in long division. Every multiply-and-add step produces the next quotient coefficient. The final number in the bottom row, the remainder, is the same value you get after subtracting the last product in long division. OpenStax College Algebra 2e section 5.4 presents both methods side by side and notes that synthetic division is a shortcut, not a different algorithm. The overlap is exact: if you wrote out the long division and then erased the polynomial notation, you would be left with the synthetic table.
When Only Long Division Works: Degree-2+ Divisors Like x² + 1
Synthetic division only works for divisors of the form x − c. OpenStax College Algebra 2e section 5.4 states this limitation explicitly: the divisor must be linear. That means any quadratic divisor, x² + 1, x² − 3x + 2, even x², is out of bounds. You also cannot use synthetic division for a divisor like 2x − 3 without an extra step (dividing the quotient coefficients by 2). Paul's Online Math Notes covers this caveat; most textbooks bury it. The failure case: a student sets up a synthetic table for x² + 1, enters a meaningless c value, and gets garbage. Do not attempt it. If the divisor is quadratic or higher, switch immediately to polynomial long division.
Long Division Walkthrough for a Quadratic Divisor
Divide 2x⁴ − 3x³ + 0x² + 5x − 1 by x² − x + 2. The divisor is quadratic, so synthetic division does not apply.
Set Up the Division
Write the dividend inside the division bracket and the divisor outside. The missing x² term in the dividend is written as 0x² to keep place values aligned.
Divide Leading Terms
2x⁴ divided by x² gives 2x². This is the first term of the quotient.
Multiply and Subtract
Multiply the whole divisor by 2x²: 2x⁴ − 2x³ + 4x². Subtract this from the dividend. −3x³ − (−2x³) = −x³. 0x² − 4x² = −4x². Bring down the remaining terms: 5x − 1. The new dividend is −x³ − 4x² + 5x − 1.
Repeat the Process
Divide the new leading term −x³ by x² to get −x. Multiply the divisor by −x: −x³ + x² − 2x. Subtract: −4x² − x² = −5x². 5x − (−2x) = 7x. Bring down −1. New dividend: −5x² + 7x − 1.
Final Division
Divide −5x² by x² to get −5. Multiply divisor by −5: −5x² + 5x − 10. Subtract: 7x − 5x = 2x. −1 − (−10) = 9. The remainder is 2x + 9, which is a degree-1 polynomial, not a number. That is normal when dividing by a quadratic.
The quotient is 2x² − x − 5 with remainder 2x + 9. Written as 2x² − x − 5 + (2x + 9)/(x² − x + 2).
Speed and Error Comparison
For a polynomial with five or more terms, synthetic division reduces the arithmetic count by about 40% compared to long division. Each long-division cycle requires a division, a multiplication, and a subtraction. Synthetic division replaces the division with a bring-down step and the subtraction with an addition, both less error-prone. The most common error in synthetic division is the sign of c: when dividing by x + 2, you need c = −2, not +2. In long division, the sign error shows up as a wrong subtraction. In synthetic division, it corrupts every subsequent number. The second most common error is a missing zero coefficient. A polynomial like x³ + 1 must be written as x³ + 0x² + 0x + 1 in both methods, but the synthetic table collapses if you skip the zeros. Long division visually shows the missing term as a gap, making the error easier to catch.
| Condition | Method to Use | Why |
|---|---|---|
| Divisor is x − c (linear) | Synthetic division | Fewer steps, lower error rate for linear divisors |
| Divisor is x + c (e.g., x + 5) | Synthetic division (c = −5) | Same method; sign of c is negative of the constant term |
| Divisor is ax − b (e.g., 2x − 3) | Long division, or synthetic with extra step | Synthetic division on its own gives wrong quotient; must divide quotient coefficients by a |
| Divisor is quadratic or higher | Long division | Synthetic division does not work for degree-2+ divisors |
| Dividend has missing terms | Either, but check zeros | Insert 0 coefficients for missing powers in both methods |
| Fast check of P(c) value | Synthetic division | Remainder equals P(c); no extra steps needed |
| Full polynomial form required for review | Long division | Shows the polynomial terms, not just coefficients |
Common Questions
What is the difference between synthetic division and polynomial long division?
Synthetic division uses only the coefficients of the dividend in a compact table and works only for linear divisors of the form x − c. Long division writes the full polynomial terms and works for any divisor degree. They produce identical results for linear divisors, but synthetic division is faster with fewer arithmetic steps. OpenStax College Algebra 2e section 5.4 presents both methods and confirms they are equivalent for x − c.
When to use synthetic division vs long division?
Use synthetic division when the divisor is x − c. Use long division when the divisor is quadratic (like x² + 1), cubic, or any nonlinear polynomial, or when the divisor has a leading coefficient other than 1 (like 2x − 3) unless you are willing to adjust the quotient afterward. The rule is one check: if the divisor is not x − c, use long division.
How do I divide by a quadratic polynomial?
Use polynomial long division. Synthetic division cannot handle quadratic divisors. Set up the dividend and divisor as in regular long division. Divide the leading term of the dividend by the leading term of the divisor (x²). Multiply the whole divisor by that result, subtract, and repeat. The remainder may be a degree-1 polynomial, not a constant. Work through the walkthrough above for a concrete example.
What is the most common mistake in synthetic division?
Using the wrong sign for c. When dividing by x + 2, the root of x + 2 = 0 is −2, so c = −2. Entering +2 instead gives a completely wrong quotient and remainder. Always solve x − c = 0 to find c. The second most common mistake is forgetting to include a zero coefficient for missing terms, for example, writing x³ + 1 as 1, 0, 0, 1 in the coefficient row.
Can I check my synthetic division answer quickly?
Yes. Use the Remainder Theorem: evaluate the original polynomial P(x) at x = c. The result should equal the last number in the bottom row of the synthetic table (the remainder). If they match, your division is correct. This check takes about 10 seconds and catches sign errors and arithmetic mistakes. OpenStax College Algebra 2e section 5.5 states the Remainder Theorem and its use as a verification tool.