Synthetic division is a shortcut for dividing a polynomial by x − c. This guide walks through the steps, the tricky cases, and what the remainder tells you.
What is synthetic division?
Synthetic division is a short way to divide a polynomial by a linear divisor of the form x − c. You work with the coefficients only, multiplying and adding down columns instead of writing out full long division.
It gives the same quotient and remainder as polynomial long division, in a fraction of the writing. The catch is that it only works for linear divisors. For something like x² + 1, you need long division.
You set it up as a small table with three rows. The top row holds the coefficients, the middle row holds the products you make along the way, and the bottom row, under a line, holds the answer. The value of c sits in a box to the left.
How do you do synthetic division step by step?
Put c in a box, list the coefficients, bring down the first one, then repeat "multiply by c, add to the next column" until you run out. Here is (x³ − 2x² − 4) ÷ (x − 3).
- The divisor is x − 3, so c = 3.
- Write the coefficients in order of power: 1, −2, 0, −4. The 0 stands for the missing x term.
- Bring down the 1.
- Multiply 1 × 3 = 3. Add to −2 to get 1.
- Multiply 1 × 3 = 3. Add to 0 to get 3.
- Multiply 3 × 3 = 9. Add to −4 to get 5.
- The bottom row is 1, 1, 3 | 5. The last number is the remainder.
What do you do with missing terms in synthetic division?
Write a 0 for each power of x that doesn't appear. Every column stands for one power, so skipping one shifts all the later numbers into the wrong place.
Take (x⁴ − 3x² + 2x − 5) ÷ (x − 2). There is no x³ term, so the coefficients are 1, 0, −3, 2, −5.
How do you use synthetic division with x + c?
Rewrite x + c as x − (−c) and put −c in the box. For a divisor of x + 3, the number you use is −3.
Here is (2x³ + 3x² − 8x + 3) ÷ (x + 3). Bring down 2. Then 2 × (−3) = −6, and 3 + (−6) = −3. Next, −3 × (−3) = 9, and −8 + 9 = 1. Last, 1 × (−3) = −3, and 3 + (−3) = 0.
What does the remainder theorem say?
The remainder theorem says that when you divide f(x) by x − c, the remainder equals f(c). So synthetic division is also a quick way to evaluate a polynomial.
In the first example, f(x) = x³ − 2x² − 4 and the remainder was 5. Check it directly: f(3) = 27 − 18 − 4 = 5. The same holds for the missing-term example, where f(2) = 16 − 12 + 4 − 5 = 3.
This is handy when the polynomial is long or the input is awkward. Some textbooks call it synthetic substitution. The bottom row builds the value one small multiplication at a time, so you never have to work out a large power on its own.
How does synthetic division show a factor?
If the remainder is 0, then x − c is a factor of the polynomial and c is a zero. That's the factor theorem, and it's the main reason synthetic division shows up in algebra 2 and precalculus.
The x + 3 example ended in 0, so x + 3 is a factor of 2x³ + 3x² − 8x + 3. The quotient 2x² − 3x + 1 factors as (2x − 1)(x − 1). That gives the full factorization and all three zeros: −3, 1/2, and 1.
How do you do synthetic division by 2x − 1?
Divide by x − 1/2 using c = 1/2, then divide every quotient coefficient by 2. The remainder stays as it is.
Try (2x³ + x² − 5x + 2) ÷ (2x − 1). With c = 1/2: bring down 2. Then 2 × 1/2 = 1, and 1 + 1 = 2. Next, 2 × 1/2 = 1, and −5 + 1 = −4. Last, −4 × 1/2 = −2, and 2 + (−2) = 0.
How do you use synthetic division to find all the zeros?
Test a likely zero with synthetic division. When the remainder is 0, the quotient is a smaller polynomial, and you keep going until you reach a quadratic you can factor or solve.
Take x³ − x² − 14x + 24. Any rational zero has to divide 24, so small values like 1, 2, and 3 are good first guesses. Testing c = 2 gives a bottom row of 1, 1, −12 | 0. The remainder is 0, so x − 2 is a factor.
Is synthetic division faster than long division?
For a linear divisor, yes. Synthetic division drops the variables and replaces each subtract-and-bring-down step with one multiplication and one addition.
You can see the link in the numbers. In long division of x³ − 2x² − 4 by x − 3, the first step subtracts x³ − 3x² and leaves x². Synthetic division gets that same 1 by multiplying 1 × 3 and adding it to −2. The sign change is built into using 3 instead of −3, which is why you add instead of subtract.
Long division still has a job. It handles any divisor, including x² + 1 or x² − 3x + 2, where synthetic division doesn't apply. The polynomial long division calculator shows the full layout if you want to compare the two methods on the same problem.
What are common synthetic division mistakes?
The method is short, so one small slip changes everything after it. Check for these.
- Using +3 for a divisor of x + 3. The box needs −3.
- Leaving out a 0 for a missing power.
- Subtracting in the columns. Synthetic division adds; long division is the one that subtracts.
- Starting the quotient at the wrong degree. It is always one less than the original.
- Forgetting to divide the quotient by 2 (or by a) when the divisor is ax − b.
Frequently asked questions
When can you use synthetic division?
Use it when the divisor is linear, like x − 4, x + 2, or 3x − 1. For a divisor of degree 2 or higher, such as x² − 1, use polynomial long division instead.
What does the last number in synthetic division mean?
The last number is the remainder. By the remainder theorem it also equals f(c), the value of the polynomial at x = c. If it is 0, x − c is a factor.
How do you write the answer to synthetic division?
Write the quotient, then add the remainder over the divisor. For (x³ − 2x² − 4) ÷ (x − 3), the answer is x² + x + 3 + 5/(x − 3).
Is synthetic division the same as long division?
It gives the same quotient and remainder. Synthetic division just drops the variables and the repeated subtraction, so it is quicker for linear divisors.
Can you use a fraction in synthetic division?
Yes. For a divisor of x − 2/3, put 2/3 in the box and work with fractions. For 3x − 2, use 2/3 and then divide the quotient by 3, the same way the 2x − 1 example divides by 2.
Why do you put a zero for missing terms?
Each column stands for one power of x. A zero keeps the empty power in its place so the multiplying and adding line up with the right terms.